 0.1: Classify the following real numbers as rational or irrational.
 0.2: Classify the following real numbers as rational or irrational.
 0.3: Classify the following real numbers as rational or irrational.
 0.4: Classify the following real numbers as rational or irrational.
 0.5: Classify the following real numbers as rational or irrational.
 0.6: Classify the following real numbers as rational or irrational.
 0.7: Classify the following real numbers as rational or irrational.
 0.8: Classify the following real numbers as rational or irrational.
 0.9: Approximate the real number to three decimal places by (a) rounding...
 0.10: Approximate the real number to three decimal places by (a) rounding...
 0.11: Approximate the real number to three decimal places by (a) rounding...
 0.12: Approximate the real number to three decimal places by (a) rounding...
 0.13: Approximate the real number to three decimal places by (a) rounding...
 0.14: Approximate the real number to three decimal places by (a) rounding...
 0.15: Approximate the real number to three decimal places by (a) rounding...
 0.16: Approximate the real number to three decimal places by (a) rounding...
 0.17: Perform the indicated operations in the correct order.
 0.18: Perform the indicated operations in the correct order.
 0.19: Perform the indicated operations in the correct order.
 0.20: Perform the indicated operations in the correct order.
 0.21: Perform the indicated operations in the correct order.
 0.22: Perform the indicated operations in the correct order.
 0.23: Perform the indicated operations in the correct order.
 0.24: Perform the indicated operations in the correct order.
 0.25: Perform the indicated operations in the correct order.
 0.26: Perform the indicated operations in the correct order.
 0.27: Perform the indicated operations in the correct order.
 0.28: Perform the indicated operations in the correct order.
 0.29: Perform the indicated operations in the correct order.
 0.30: Perform the indicated operations in the correct order.
 0.31: Perform the indicated operations in the correct order.
 0.32: Perform the indicated operations in the correct order.
 0.33: Perform the indicated operations in the correct order.
 0.34: Perform the indicated operations in the correct order.
 0.35: Perform the indicated operations in the correct order.
 0.36: Perform the indicated operations in the correct order.
 0.37: Perform the indicated operations in the correct order.
 0.38: Perform the indicated operations in the correct order.
 0.39: Perform the indicated operations in the correct order.
 0.40: Perform the indicated operations in the correct order.
 0.41: Write as a single fraction and simplify.
 0.42: Write as a single fraction and simplify.
 0.43: Write as a single fraction and simplify.
 0.44: Write as a single fraction and simplify.
 0.45: Write as a single fraction and simplify.
 0.46: Write as a single fraction and simplify.
 0.47: Write as a single fraction and simplify.
 0.48: Write as a single fraction and simplify.
 0.49: Write as a single fraction and simplify.
 0.50: Write as a single fraction and simplify.
 0.51: Write as a single fraction and simplify.
 0.52: Write as a single fraction and simplify.
 0.53: Write as a single fraction and simplify.
 0.54: Write as a single fraction and simplify.
 0.55: Write as a single fraction and simplify.
 0.56: Write as a single fraction and simplify.
 0.57: Perform the indicated operation and simplify, if possible.
 0.58: Perform the indicated operation and simplify, if possible.
 0.59: Perform the indicated operation and simplify, if possible.
 0.60: Perform the indicated operation and simplify, if possible.
 0.61: Perform the indicated operation and simplify, if possible.
 0.62: Perform the indicated operation and simplify, if possible.
 0.63: Perform the indicated operation and simplify, if possible.
 0.64: Perform the indicated operation and simplify, if possible.
 0.65: Perform the indicated operation and simplify, if possible.
 0.66: Perform the indicated operation and simplify, if possible.
 0.67: Perform the indicated operation and simplify, if possible.
 0.68: Perform the indicated operation and simplify, if possible.
 0.69: Evaluate the algebraic expression for the specified values.
 0.70: Evaluate the algebraic expression for the specified values.
 0.71: Evaluate the algebraic expression for the specified values.
 0.72: Evaluate the algebraic expression for the specified values.
 0.73: On December 16, 2007, the United States debt was estimated at $9,17...
 0.74: On December 16, 2007, the United States debt was estimated at $9,17...
 0.75: On December 16, 2007, the United States debt was estimated at $9,17...
 0.76: On December 16, 2007, the United States debt was estimated at $9,17...
 0.77: Explain the mistake that is made. Round 13.2749 to two decimal plac...
 0.78: Explain the mistake that is made. Simplify the expression .Solution...
 0.79: Explain the mistake that is made. Simplify the expression 3(x 5) 2(...
 0.80: Explain the mistake that is made.Simplify the expression 3(x 2) (1 ...
 0.81: Determine whether each of the following statements is true or false...
 0.82: Determine whether each of the following statements is true or false...
 0.83: Determine whether each of the following statements is true or false...
 0.84: Determine whether each of the following statements is true or false...
 0.85: What restrictions are there on x for the following to be true:
 0.86: What restrictions are there on x for the following to be true:
 0.87: Simplify the expressions.
 0.88: Simplify the expressions.
 0.89: Use your calculator to evaluate . Does the answerappear to be a rat...
 0.90: Use your calculator to evaluate . Does the answerappear to be a rat...
 0.91: Use your calculator to evaluate . Does the answerappear to be a rat...
 0.92: Use your calculator to evaluate . Does the answerappear to be a rat...
 0.1: Evaluate each expression.
 0.2: Evaluate each expression.
 0.3: Evaluate each expression.
 0.4: Evaluate each expression.
 0.5: Evaluate each expression.
 0.6: Evaluate each expression.
 0.7: Evaluate each expression.
 0.8: Evaluate each expression.
 0.9: Evaluate each expression.
 0.10: Evaluate each expression.
 0.11: Evaluate each expression.
 0.12: Evaluate each expression.
 0.13: Evaluate each expression.
 0.14: Evaluate each expression.
 0.15: Evaluate each expression.
 0.16: Evaluate each expression.
 0.17: Evaluate each expression.
 0.18: Evaluate each expression.
 0.19: Evaluate each expression.
 0.20: Evaluate each expression.
 0.21: Simplify and write the resulting expression with only positive expo...
 0.22: Simplify and write the resulting expression with only positive expo...
 0.23: Simplify and write the resulting expression with only positive expo...
 0.24: Simplify and write the resulting expression with only positive expo...
 0.25: Simplify and write the resulting expression with only positive expo...
 0.26: Simplify and write the resulting expression with only positive expo...
 0.27: Simplify and write the resulting expression with only positive expo...
 0.28: Simplify and write the resulting expression with only positive expo...
 0.29: Simplify and write the resulting expression with only positive expo...
 0.30: Simplify and write the resulting expression with only positive expo...
 0.31: Simplify and write the resulting expression with only positive expo...
 0.32: Simplify and write the resulting expression with only positive expo...
 0.33: Simplify and write the resulting expression with only positive expo...
 0.34: Simplify and write the resulting expression with only positive expo...
 0.35: Simplify and write the resulting expression with only positive expo...
 0.36: Simplify and write the resulting expression with only positive expo...
 0.37: Simplify and write the resulting expression with only positive expo...
 0.38: Simplify and write the resulting expression with only positive expo...
 0.39: Simplify and write the resulting expression with only positive expo...
 0.40: Simplify and write the resulting expression with only positive expo...
 0.41: Simplify and write the resulting expression with only positive expo...
 0.42: Simplify and write the resulting expression with only positive expo...
 0.43: Simplify and write the resulting expression with only positive expo...
 0.44: Simplify and write the resulting expression with only positive expo...
 0.45: Simplify and write the resulting expression with only positive expo...
 0.46: Simplify and write the resulting expression with only positive expo...
 0.47: Simplify and write the resulting expression with only positive expo...
 0.48: Simplify and write the resulting expression with only positive expo...
 0.49: Simplify and write the resulting expression with only positive expo...
 0.50: Simplify and write the resulting expression with only positive expo...
 0.51: Write 28 163 (64) as
 0.52: Write 39 815 (9) asa power of 2 : 2? a power of 3 : 3?
 0.53: Express the given number in scientific notation.
 0.54: Express the given number in scientific notation.
 0.55: Express the given number in scientific notation.
 0.56: Express the given number in scientific notation.
 0.57: Express the given number in scientific notation.
 0.58: Express the given number in scientific notation.
 0.59: Express the given number in scientific notation.
 0.60: Express the given number in scientific notation.
 0.61: Write the number as a decimal.
 0.62: Write the number as a decimal.
 0.63: Write the number as a decimal.
 0.64: Write the number as a decimal.
 0.65: Write the number as a decimal.
 0.66: Write the number as a decimal.
 0.67: Refer to the following:It is estimated that there are currently cel...
 0.68: Refer to the following:It is estimated that there are currently cel...
 0.69: Astronomy. The distance from Earth to Mars on a particularday can b...
 0.70: Astronomy. The distance from Mars to the Sun on aparticular day can...
 0.71: Lasers. The wavelength of a typical laser used forcommunication sys...
 0.72: Lasers. A rubyred laser has a wavelength of 694 nanometers(or 6.93...
 0.73: Explain the mistake that is made.Simplify (2y3)(3x2 y2).Group like ...
 0.74: Explain the mistake that is made.Simplify (2xy2)3.Eliminate the par...
 0.75: Explain the mistake that is made.Simplify (2xy3)2(5x2 y)2.Apply the...
 0.76: Explain the mistake that is made.Simplify .Group like factors.Use t...
 0.77: Determine whether each of the followingstatements is true or false....
 0.78: Determine whether each of the followingstatements is true or false....
 0.79: Determine whether each of the followingstatements is true or false....
 0.80: Determine whether each of the followingstatements is true or false....
 0.81: Simplify
 0.82: Simplify
 0.83: Evaluate the expression for the given value.a2 2ab for a 2, b 3
 0.84: Evaluate the expression for the given value.2a3 7a2 for a 4
 0.85: Evaluate the expression for the given value.16t2 100t for t
 0.86: Evaluate the expression for the given value.
 0.87: The Earths population is approximately 6.6 109 people,and there are...
 0.88: The population of the United States is approximately3.0 108 people,...
 0.89: Evaluate: . Express your answer inboth scientific and decimal notat...
 0.90: Evaluate: . Express your answer inboth scientific and decimal notat...
 0.91: Scientific calculators have an EXP button that is used forscientifi...
 0.92: Scientific calculators have an EXP button that is used forscientifi...
 0.93: Use a graphing utility or scientificcalculator to evaluate the expr...
 0.94: Use a graphing utility or scientificcalculator to evaluate the expr...
 0.1: Write the polynomial in standard form and state the degree of the p...
 0.2: Write the polynomial in standard form and state the degree of the p...
 0.3: Write the polynomial in standard form and state the degree of the p...
 0.4: Write the polynomial in standard form and state the degree of the p...
 0.5: Write the polynomial in standard form and state the degree of the p...
 0.6: Write the polynomial in standard form and state the degree of the p...
 0.7: Write the polynomial in standard form and state the degree of the p...
 0.8: Write the polynomial in standard form and state the degree of the p...
 0.9: Add or subtract the polynomials, gather like terms, and write the s...
 0.10: Add or subtract the polynomials, gather like terms, and write the s...
 0.11: Add or subtract the polynomials, gather like terms, and write the s...
 0.12: Add or subtract the polynomials, gather like terms, and write the s...
 0.13: Add or subtract the polynomials, gather like terms, and write the s...
 0.14: Add or subtract the polynomials, gather like terms, and write the s...
 0.15: Add or subtract the polynomials, gather like terms, and write the s...
 0.16: Add or subtract the polynomials, gather like terms, and write the s...
 0.17: Add or subtract the polynomials, gather like terms, and write the s...
 0.18: Add or subtract the polynomials, gather like terms, and write the s...
 0.19: Add or subtract the polynomials, gather like terms, and write the s...
 0.20: Add or subtract the polynomials, gather like terms, and write the s...
 0.21: Add or subtract the polynomials, gather like terms, and write the s...
 0.22: Add or subtract the polynomials, gather like terms, and write the s...
 0.23: Add or subtract the polynomials, gather like terms, and write the s...
 0.24: Add or subtract the polynomials, gather like terms, and write the s...
 0.25: Multiply the polynomials and write the expressions in standard form...
 0.26: Multiply the polynomials and write the expressions in standard form...
 0.27: Multiply the polynomials and write the expressions in standard form...
 0.28: Multiply the polynomials and write the expressions in standard form...
 0.29: Multiply the polynomials and write the expressions in standard form...
 0.30: Multiply the polynomials and write the expressions in standard form. .
 0.31: Multiply the polynomials and write the expressions in standard form...
 0.32: Multiply the polynomials and write the expressions in standard form...
 0.33: Multiply the polynomials and write the expressions in standard form...
 0.34: Multiply the polynomials and write the expressions in standard form...
 0.35: Multiply the polynomials and write the expressions in standard form...
 0.36: Multiply the polynomials and write the expressions in standard form...
 0.37: Multiply the polynomials and write the expressions in standard form...
 0.38: Multiply the polynomials and write the expressions in standard form...
 0.39: Multiply the polynomials and write the expressions in standard form...
 0.40: Multiply the polynomials and write the expressions in standard form...
 0.41: Multiply the polynomials and write the expressions in standard form...
 0.42: Multiply the polynomials and write the expressions in standard form...
 0.43: Multiply the polynomials and write the expressions in standard form...
 0.44: Multiply the polynomials and write the expressions in standard form...
 0.45: Multiply the polynomials and write the expressions in standard form...
 0.46: Multiply the polynomials and write the expressions in standard form...
 0.47: Multiply the polynomials and write the expressions in standard form...
 0.48: Multiply the polynomials and write the expressions in standard form...
 0.49: Multiply the polynomials and write the expressions in standard form...
 0.50: Multiply the polynomials and write the expressions in standard form...
 0.51: Multiply the polynomials and write the expressions in standard form...
 0.52: Multiply the polynomials and write the expressions in standard form...
 0.53: Multiply the polynomials and write the expressions in standard form...
 0.54: Multiply the polynomials and write the expressions in standard form...
 0.55: Multiply the polynomials and write the expressions in standard form...
 0.56: Multiply the polynomials and write the expressions in standard form...
 0.57: Multiply the polynomials and write the expressions in standard form...
 0.58: Multiply the polynomials and write the expressions in standard form...
 0.59: Multiply the polynomials and write the expressions in standard form...
 0.60: Multiply the polynomials and write the expressions in standard form...
 0.61: Multiply the polynomials and write the expressions in standard form...
 0.62: Multiply the polynomials and write the expressions in standard form...
 0.63: Multiply the polynomials and write the expressions in standard form...
 0.64: Multiply the polynomials and write the expressions in standard form...
 0.65: Prot is equal to revenue minus cost: P R C. Prot. Donna decides to ...
 0.66: Prot is equal to revenue minus cost: P R C.Prot. Calculators are so...
 0.67: Prot is equal to revenue minus cost: P R C.Prot.Iftherevenueassocia...
 0.68: Prot is equal to revenue minus cost: P R C.Prot.A business sells a ...
 0.69: Volume of a Box. A rectangular sheet of cardboard is to be used in ...
 0.70: Volume of a Box. Suppose a box is to be constructed from a square p...
 0.71: Geometry. Suppose a running track is constructed of a rectangular p...
 0.72: Geometry.A right circular cylinder whose radius is r and whose heig...
 0.73: Engineering. The force of an electrical eld is given by theequation...
 0.74: Engineering. If a football (or other projectile) is thrown upward, ...
 0.75: Explain the mistake that is made.Subtract and simplify Solution: El...
 0.76: Explain the mistake that is made.Simplify (2 x)2. Solution: Write t...
 0.77: Determine whether each of the following statements is true or false...
 0.78: Determine whether each of the following statements is true or false...
 0.79: Determine whether each of the following statements is true or false...
 0.80: Determine whether each of the following statements is true or false...
 0.81: Let m and n be real numbers and . What degree is the product of a p...
 0.82: Let m and n be real numbers and . What degree is the sum of a polyn...
 0.83: Perform the indicated operations and simplify. (7x 4y2)2 (7x 4y2)2
 0.84: Perform the indicated operations and simplify. (3x 5y2)2(3x 5y2)2
 0.85: Perform the indicated operations and simplify. (x a)(x2 ax a2)
 0.86: Perform the indicated operations and simplify. (x a)(x2 ax a2)
 0.87: Use a graphing utility to plot the graphs of the three expressions ...
 0.88: Use a graphing utility to plot the graphs of the three expressions ...
 0.1: Factor each expression. Start by nding the greatest common factor (...
 0.2: Factor each expression. Start by nding the greatest common factor (...
 0.3: Factor each expression. Start by nding the greatest common factor (...
 0.4: Factor each expression. Start by nding the greatest common factor (...
 0.5: Factor each expression. Start by nding the greatest common factor (...
 0.6: Factor each expression. Start by nding the greatest common factor (...
 0.7: Factor each expression. Start by nding the greatest common factor (...
 0.8: Factor each expression. Start by nding the greatest common factor (...
 0.9: Factor each expression. Start by nding the greatest common factor (...
 0.10: Factor each expression. Start by nding the greatest common factor (...
 0.11: Factor each expression. Start by nding the greatest common factor (...
 0.12: Factor each expression. Start by nding the greatest common factor (...
 0.13: Factor the difference of two squares. x2 9
 0.14: Factor the difference of two squares. x2 25
 0.15: Factor the difference of two squares. 4x2 9
 0.16: Factor the difference of two squares. 1 x4
 0.17: Factor the difference of two squares. 2x2 98
 0.18: Factor the difference of two squares. 144 81y2
 0.19: Factor the difference of two squares. 225x2 169y2
 0.20: Factor the difference of two squares. 121y2 49x2
 0.21: Factor the perfect squares. x2 8x 16
 0.22: Factor the perfect squares. y2 10y 25
 0.23: Factor the perfect squares. x4 4x2 4
 0.24: Factor the perfect squares. 1 6y 9y2
 0.25: Factor the perfect squares. 4x2 12xy 9y2
 0.26: Factor the perfect squares. x2 6xy 9y2
 0.27: Factor the perfect squares. 9 6x x2
 0.28: Factor the perfect squares. 25x2 20xy 4y2
 0.29: Factor the perfect squares. x4 2x2 1
 0.30: Factor the perfect squares. x6 6x3 9
 0.31: Factor the perfect squares. p2 2pq q2
 0.32: Factor the perfect squares. p2 2pq q2
 0.33: Factor the sum or difference of two cubes. t3 27
 0.34: Factor the sum or difference of two cubes. z3 64
 0.35: Factor the sum or difference of two cubes. y3 64
 0.36: Factor the sum or difference of two cubes. x3 1
 0.37: Factor the sum or difference of two cubes. 8 x3
 0.38: Factor the sum or difference of two cubes. 27 y3
 0.39: Factor the sum or difference of two cubes. y3 125
 0.40: Factor the sum or difference of two cubes. 64x x4
 0.41: Factor the sum or difference of two cubes. 27 x3
 0.42: Factor the sum or difference of two cubes. 216x3 y3
 0.43: Factor each trinomial into a product of two binomials. x2 6x 5
 0.44: Factor each trinomial into a product of two binomials. t2 5t 6
 0.45: Factor each trinomial into a product of two binomials. y2 2y 3
 0.46: Factor each trinomial into a product of two binomials. y2 3y 10
 0.47: Factor each trinomial into a product of two binomials. 2y2 5y 3
 0.48: Factor each trinomial into a product of two binomials. 2z2 4z 6
 0.49: Factor each trinomial into a product of two binomials. 3t2 7t 2
 0.50: Factor each trinomial into a product of two binomials. 4x2 2x 12
 0.51: Factor each trinomial into a product of two binomials. 6t2 t 2
 0.52: Factor each trinomial into a product of two binomials. 6x2 17x 10
 0.53: Factor by grouping. x3 3x2 2x 6
 0.54: Factor by grouping. x5 5x3 3x2 15
 0.55: Factor by grouping. a4 2a3 8a 16
 0.56: Factor by grouping. x4 3x3 x 3
 0.57: Factor by grouping. 3xy 5rx 10rs 6sy
 0.58: Factor by grouping. 6x2 10x 3x 5
 0.59: Factor by grouping. 20x2 8xy 5xy 2y2
 0.60: Factor by grouping. 9x5 a2x3 9x2 a2
 0.61: Factor each of the polynomials completely,if possible. If the polyn...
 0.62: Factor each of the polynomials completely,if possible. If the polyn...
 0.63: Factor each of the polynomials completely,if possible. If the polyn...
 0.64: Factor each of the polynomials completely,if possible. If the polyn...
 0.65: Factor each of the polynomials completely,if possible. If the polyn...
 0.66: Factor each of the polynomials completely,if possible. If the polyn...
 0.67: Factor each of the polynomials completely,if possible. If the polyn...
 0.68: Factor each of the polynomials completely,if possible. If the polyn...
 0.69: Factor each of the polynomials completely,if possible. If the polyn...
 0.70: Factor each of the polynomials completely,if possible. If the polyn...
 0.71: Factor each of the polynomials completely,if possible. If the polyn...
 0.72: Factor each of the polynomials completely,if possible. If the polyn...
 0.73: Factor each of the polynomials completely,if possible. If the polyn...
 0.74: Factor each of the polynomials completely,if possible. If the polyn...
 0.75: Factor each of the polynomials completely,if possible. If the polyn...
 0.76: Factor each of the polynomials completely,if possible. If the polyn...
 0.77: Factor each of the polynomials completely,if possible. If the polyn...
 0.78: Factor each of the polynomials completely,if possible. If the polyn...
 0.79: Factor each of the polynomials completely,if possible. If the polyn...
 0.80: Factor each of the polynomials completely,if possible. If the polyn...
 0.81: Factor each of the polynomials completely,if possible. If the polyn...
 0.82: Factor each of the polynomials completely,if possible. If the polyn...
 0.83: Factor each of the polynomials completely,if possible. If the polyn...
 0.84: Factor each of the polynomials completely,if possible. If the polyn...
 0.85: Factor each of the polynomials completely,if possible. If the polyn...
 0.86: Factor each of the polynomials completely,if possible. If the polyn...
 0.87: Factor each of the polynomials completely,if possible. If the polyn...
 0.88: Factor each of the polynomials completely,if possible. If the polyn...
 0.89: Factor each of the polynomials completely,if possible. If the polyn...
 0.90: Factor each of the polynomials completely,if possible. If the polyn...
 0.91: Factor each of the polynomials completely,if possible. If the polyn...
 0.92: Factor each of the polynomials completely,if possible. If the polyn...
 0.93: Geometry.A rectangle has a length of 2x 4 and a width of x. Express...
 0.94: Geometry. The volume of a box is given by the expression x3 7x2 12x...
 0.95: Business. The prot of a business is given by the expression P 2x2 1...
 0.96: Business. The breakeven point for a company is given by solving th...
 0.97: Engineering. The height of a projectile is given by the equation s ...
 0.98: Engineering. The electrical eld at a point P between twocharges is ...
 0.99: Explain the mistake that is made. Factor x3 x2 9x 9. Solution: Grou...
 0.100: Explain the mistake that is made.Factor 4x2 12x 40. Solution: Facto...
 0.101: Determine whether each of the following statements is true or false...
 0.102: Determine whether each of the following statements is true or false...
 0.103: Determine whether each of the following statements is true or false...
 0.104: Determine whether each of the following statements is true or false...
 0.105: Factor a2n b2n completely, assuming a, b, and n are positive integers.
 0.106: Find all the values of c such that the trinomial x2 cx 14 can be fa...
 0.107: Use a graphing utility to plot the graphs of the three expressions ...
 0.108: Use a graphing utility to plot the graphs of the three expressions ...
 0.1: State any real numbers that must be excluded from the domain of eac...
 0.2: State any real numbers that must be excluded from the domain of eac...
 0.3: State any real numbers that must be excluded from the domain of eac...
 0.4: State any real numbers that must be excluded from the domain of eac...
 0.5: State any real numbers that must be excluded from the domain of eac...
 0.6: State any real numbers that must be excluded from the domain of eac...
 0.7: State any real numbers that must be excluded from the domain of eac...
 0.8: State any real numbers that must be excluded from the domain of eac...
 0.9: State any real numbers that must be excluded from the domain of eac...
 0.10: State any real numbers that must be excluded from the domain of eac...
 0.11: Reduce the rational expression to lowest terms and state any real n...
 0.12: Reduce the rational expression to lowest terms and state any real n...
 0.13: Reduce the rational expression to lowest terms and state any real n...
 0.14: Reduce the rational expression to lowest terms and state any real n...
 0.15: Reduce the rational expression to lowest terms and state any real n...
 0.16: Reduce the rational expression to lowest terms and state any real n...
 0.17: Reduce the rational expression to lowest terms and state any real n...
 0.18: Reduce the rational expression to lowest terms and state any real n...
 0.19: Reduce the rational expression to lowest terms and state any real n...
 0.20: Reduce the rational expression to lowest terms and state any real n...
 0.21: Reduce the rational expression to lowest terms and state any real n...
 0.22: Reduce the rational expression to lowest terms and state any real n...
 0.23: Reduce the rational expression to lowest terms and state any real n...
 0.24: Reduce the rational expression to lowest terms and state any real n...
 0.25: Reduce the rational expression to lowest terms and state any real n...
 0.26: Reduce the rational expression to lowest terms and state any real n...
 0.27: Reduce the rational expression to lowest terms and state any real n...
 0.28: Reduce the rational expression to lowest terms and state any real n...
 0.29: Reduce the rational expression to lowest terms and state any real n...
 0.30: Reduce the rational expression to lowest terms and state any real n...
 0.31: Multiply the rational expressions and simplify. State any real numb...
 0.32: Multiply the rational expressions and simplify. State any real numb...
 0.33: Multiply the rational expressions and simplify. State any real numb...
 0.34: Multiply the rational expressions and simplify. State any real numb...
 0.35: Multiply the rational expressions and simplify. State any real numb...
 0.36: Multiply the rational expressions and simplify. State any real numb...
 0.37: Multiply the rational expressions and simplify. State any real numb...
 0.38: Multiply the rational expressions and simplify. State any real numb...
 0.39: Multiply the rational expressions and simplify. State any real numb...
 0.40: Multiply the rational expressions and simplify. State any real numb...
 0.41: Multiply the rational expressions and simplify. State any real numb...
 0.42: Multiply the rational expressions and simplify. State any real numb...
 0.43: Multiply the rational expressions and simplify. State any real numb...
 0.44: Multiply the rational expressions and simplify. State any real numb...
 0.45: Multiply the rational expressions and simplify. State any real numb...
 0.46: Multiply the rational expressions and simplify. State any real numb...
 0.47: Multiply the rational expressions and simplify. State any real numb...
 0.48: Multiply the rational expressions and simplify. State any real numb...
 0.49: Divide the rational expressions and simplify. State any real number...
 0.50: Divide the rational expressions and simplify. State any real number...
 0.51: Divide the rational expressions and simplify. State any real number...
 0.52: Divide the rational expressions and simplify. State any real number...
 0.53: Divide the rational expressions and simplify. State any real number...
 0.54: Divide the rational expressions and simplify. State any real number...
 0.55: Divide the rational expressions and simplify. State any real number...
 0.56: Divide the rational expressions and simplify. State any real number...
 0.57: Divide the rational expressions and simplify. State any real number...
 0.58: Divide the rational expressions and simplify. State any real number...
 0.59: Divide the rational expressions and simplify. State any real number...
 0.60: Divide the rational expressions and simplify. State any real number...
 0.61: Divide the rational expressions and simplify. State any real number...
 0.62: Divide the rational expressions and simplify. State any real number...
 0.63: Divide the rational expressions and simplify. State any real number...
 0.64: Divide the rational expressions and simplify. State any real number...
 0.65: Divide the rational expressions and simplify. State any real number...
 0.66: Divide the rational expressions and simplify. State any real number...
 0.67: Add or subtract the rational expression and simplify. State any rea...
 0.68: Add or subtract the rational expression and simplify. State any rea...
 0.69: Add or subtract the rational expression and simplify. State any rea...
 0.70: Add or subtract the rational expression and simplify. State any rea...
 0.71: Add or subtract the rational expression and simplify. State any rea...
 0.72: Add or subtract the rational expression and simplify. State any rea...
 0.73: Add or subtract the rational expression and simplify. State any rea...
 0.74: Add or subtract the rational expression and simplify. State any rea...
 0.75: Add or subtract the rational expression and simplify. State any rea...
 0.76: Add or subtract the rational expression and simplify. State any rea...
 0.77: Add or subtract the rational expression and simplify. State any rea...
 0.78: Add or subtract the rational expression and simplify. State any rea...
 0.79: Add or subtract the rational expression and simplify. State any rea...
 0.80: Add or subtract the rational expression and simplify. State any rea...
 0.81: Add or subtract the rational expression and simplify. State any rea...
 0.82: Add or subtract the rational expression and simplify. State any rea...
 0.83: Simplify the complex rational expressions. State any real numbers t...
 0.84: Simplify the complex rational expressions. State any real numbers t...
 0.85: Simplify the complex rational expressions. State any real numbers t...
 0.86: Simplify the complex rational expressions. State any real numbers t...
 0.87: Simplify the complex rational expressions. State any real numbers t...
 0.88: Simplify the complex rational expressions. State any real numbers t...
 0.89: Simplify the complex rational expressions. State any real numbers t...
 0.90: Simplify the complex rational expressions. State any real numbers t...
 0.91: Finance. The amount of payment made on a loan is given by the formu...
 0.92: Finance. Use the formula to calculate the amount your monthly payme...
 0.93: Circuits. If two resistors are connected in parallel, the combined ...
 0.94: Optics. The focal length of a lens can be calculated by applying th...
 0.95: Explain the mistake that is made. Simplify .Solution:Factor the num...
 0.96: Explain the mistake that is made.Simplify .Solution:Cancel the comm...
 0.97: Determine whether each of the statements is true or false.
 0.98: Determine whether each of the statements is true or false.
 0.99: Determine whether each of the statements is true or false.When addi...
 0.100: Determine whether each of the statements is true or false.
 0.101: Perform the operation and simplify (remember to state domain restri...
 0.102: Write the numerator as the product of two binomials. Divide out any...
 0.103: Utilizing a graphing technology, plot the expression. Zoom in near ...
 0.104: Utilizing a graphing technology, plot the expression. Zoom in near ...
 0.105: Utilizing a graphing technology, plot the expression. Zoom in near ...
 0.106: Utilizing a graphing technology, plot the expression. Zoom in near ...
 0.1: Evaluate each expression or state that it is not a real number.
 0.2: Evaluate each expression or state that it is not a real number.
 0.3: Evaluate each expression or state that it is not a real number.
 0.4: Evaluate each expression or state that it is not a real number.
 0.5: Evaluate each expression or state that it is not a real number.
 0.6: Evaluate each expression or state that it is not a real number.
 0.7: Evaluate each expression or state that it is not a real number.
 0.8: Evaluate each expression or state that it is not a real number.
 0.9: Evaluate each expression or state that it is not a real number.
 0.10: Evaluate each expression or state that it is not a real number.
 0.11: Evaluate each expression or state that it is not a real number.
 0.12: Evaluate each expression or state that it is not a real number.
 0.13: Evaluate each expression or state that it is not a real number.
 0.14: Evaluate each expression or state that it is not a real number.
 0.15: Evaluate each expression or state that it is not a real number.
 0.16: Evaluate each expression or state that it is not a real number.
 0.17: Evaluate each expression or state that it is not a real number.
 0.18: Evaluate each expression or state that it is not a real number.
 0.19: Evaluate each expression or state that it is not a real number.
 0.20: Evaluate each expression or state that it is not a real number.
 0.21: Evaluate each expression or state that it is not a real number.
 0.22: Evaluate each expression or state that it is not a real number.
 0.23: Evaluate each expression or state that it is not a real number.
 0.24: Evaluate each expression or state that it is not a real number.
 0.25: Simplify (if possible) the radical expressions.
 0.26: Simplify (if possible) the radical expressions.
 0.27: Simplify (if possible) the radical expressions.
 0.28: Simplify (if possible) the radical expressions.
 0.29: Simplify (if possible) the radical expressions.
 0.30: Simplify (if possible) the radical expressions.
 0.31: Simplify (if possible) the radical expressions.
 0.32: Simplify (if possible) the radical expressions.
 0.33: Simplify (if possible) the radical expressions.
 0.34: Simplify (if possible) the radical expressions.
 0.35: Simplify (if possible) the radical expressions.
 0.36: Simplify (if possible) the radical expressions.
 0.37: Simplify (if possible) the radical expressions.
 0.38: Simplify (if possible) the radical expressions.
 0.39: Simplify (if possible) the radical expressions.
 0.40: Simplify (if possible) the radical expressions.
 0.41: Rationalize the denominators.
 0.42: Rationalize the denominators.
 0.43: Rationalize the denominators.
 0.44: Rationalize the denominators.
 0.45: Rationalize the denominators.
 0.46: Rationalize the denominators.
 0.47: Rationalize the denominators.
 0.48: Rationalize the denominators.
 0.49: Rationalize the denominators.
 0.50: Rationalize the denominators.
 0.51: Rationalize the denominators.
 0.52: Rationalize the denominators.
 0.53: Rationalize the denominators.
 0.54: Rationalize the denominators.
 0.55: Rationalize the denominators.
 0.56: Rationalize the denominators.
 0.57: Simplify by applying the properties of rational exponents. Express ...
 0.58: Simplify by applying the properties of rational exponents. Express ...
 0.59: Simplify by applying the properties of rational exponents. Express ...
 0.60: Simplify by applying the properties of rational exponents. Express ...
 0.61: Simplify by applying the properties of rational exponents. Express ...
 0.62: Simplify by applying the properties of rational exponents. Express ...
 0.63: Simplify by applying the properties of rational exponents. Express ...
 0.64: Simplify by applying the properties of rational exponents. Express ...
 0.65: Factor each expression completely. x7/3 x4/3 2x1/3
 0.66: Factor each expression completely. 8x1/4 4x5/4
 0.67: Factor each expression completely. 7x3/7 14x6/7 21x10/7
 0.68: Factor each expression completely. 7x1/3 70x
 0.69: Gravity. If a penny is dropped off a building, the time it takes (s...
 0.70: Gravity. If a ball is dropped off a building, the time it takes (se...
 0.71: Keplers Law. The square of the period p(in years) of a planets orbi...
 0.72: Period of a Pendulum. The period (in seconds) of a pendulumof lengt...
 0.73: Explain the mistake that is made.Simplify Solution: Use properties ...
 0.74: Explain the mistake that is made.Simplify .Solution: Multiply numer...
 0.75: Determine whether each statement is true or false. 1121 =; 11
 0.76: Determine whether each statement is true or false. , where x is any...
 0.77: Determine whether each statement is true or false. 2a2 + b2 = 2a + 2b
 0.78: Determine whether each statement is true or false. 14 =2
 0.79: a, m, n, and k are any positive real numbers.Simplify .
 0.80: a, m, n, and k are any positive real numbers.Simplify .
 0.81: Evaluate each algebraic expression for the specied values.
 0.82: Evaluate each algebraic expression for the specied values.
 0.83: Rationalize the denominator and simplify: . 1 A1a + 1bB2
 0.84: Rationalize the denominator and simplify: . 1a + b  1a 1a +
 0.85: Use a calculator to approximate to three decimal places.
 0.86: Use a calculator to approximate to three decimal places.
 0.87: Given a. Rationalize the denominator. b. Use a graphing utility to ...
 0.88: Given a. Rationalize the denominator. b. Use a graphing utility to ...
 0.1: Write each expression as a complex number in standard form. Some ex...
 0.2: Write each expression as a complex number in standard form. Some ex...
 0.3: Write each expression as a complex number in standard form. Some ex...
 0.4: Write each expression as a complex number in standard form. Some ex...
 0.5: Write each expression as a complex number in standard form. Some ex...
 0.6: Write each expression as a complex number in standard form. Some ex...
 0.7: Write each expression as a complex number in standard form. Some ex...
 0.8: Write each expression as a complex number in standard form. Some ex...
 0.9: Write each expression as a complex number in standard form. Some ex...
 0.10: Write each expression as a complex number in standard form. Some ex...
 0.11: Write each expression as a complex number in standard form. Some ex...
 0.12: Write each expression as a complex number in standard form. Some ex...
 0.13: Perform the indicated operation, simplify, and express in standard ...
 0.14: Perform the indicated operation, simplify, and express in standard ...
 0.15: Perform the indicated operation, simplify, and express in standard ...
 0.16: Perform the indicated operation, simplify, and express in standard ...
 0.17: Perform the indicated operation, simplify, and express in standard ...
 0.18: Perform the indicated operation, simplify, and express in standard ...
 0.19: Perform the indicated operation, simplify, and express in standard ...
 0.20: Perform the indicated operation, simplify, and express in standard ...
 0.21: Perform the indicated operation, simplify, and express in standard ...
 0.22: Perform the indicated operation, simplify, and express in standard ...
 0.23: Perform the indicated operation, simplify, and express in standard ...
 0.24: Perform the indicated operation, simplify, and express in standard ...
 0.25: Perform the indicated operation, simplify, and express in standard ...
 0.26: Perform the indicated operation, simplify, and express in standard ...
 0.27: Perform the indicated operation, simplify, and express in standard ...
 0.28: Perform the indicated operation, simplify, and express in standard ...
 0.29: Perform the indicated operation, simplify, and express in standard ...
 0.30: Perform the indicated operation, simplify, and express in standard ...
 0.31: Perform the indicated operation, simplify, and express in standard ...
 0.32: Perform the indicated operation, simplify, and express in standard ...
 0.33: Perform the indicated operation, simplify, and express in standard ...
 0.34: Perform the indicated operation, simplify, and express in standard ...
 0.35: Perform the indicated operation, simplify, and express in standard ...
 0.36: Perform the indicated operation, simplify, and express in standard ...
 0.37: Perform the indicated operation, simplify, and express in standard ...
 0.38: Perform the indicated operation, simplify, and express in standard ...
 0.39: Perform the indicated operation, simplify, and express in standard ...
 0.40: Perform the indicated operation, simplify, and express in standard ...
 0.41: For each complex number z, write the complex conjugate and nd . z 4 7i
 0.42: For each complex number z, write the complex conjugate and nd . z 2 5i
 0.43: For each complex number z, write the complex conjugate and nd . z 2 3i
 0.44: For each complex number z, write the complex conjugate and nd . z 5 3i
 0.45: For each complex number z, write the complex conjugate and nd . z 6 4i
 0.46: For each complex number z, write the complex conjugate and nd . z 2 7i
 0.47: For each complex number z, write the complex conjugate and nd . z 2 6i
 0.48: For each complex number z, write the complex conjugate and nd . z 3 9i
 0.49: Write each quotient in standard form.
 0.50: Write each quotient in standard form.
 0.51: Write each quotient in standard form.
 0.52: Write each quotient in standard form.
 0.53: Write each quotient in standard form.
 0.54: Write each quotient in standard form.
 0.55: Write each quotient in standard form.
 0.56: Write each quotient in standard form.
 0.57: Write each quotient in standard form.
 0.58: Write each quotient in standard form.
 0.59: Write each quotient in standard form.
 0.60: Write each quotient in standard form.
 0.61: Write each quotient in standard form.
 0.62: Write each quotient in standard form.
 0.63: Write each quotient in standard form.
 0.64: Write each quotient in standard form.
 0.65: Simplify. i15
 0.66: Simplify. i99
 0.67: Simplify. i40
 0.68: Simplify. i18
 0.69: Simplify. (5 2i)2
 0.70: Simplify. (3 5i)2
 0.71: Simplify. (2 3i)2
 0.72: Simplify. (4 9i)2
 0.73: Simplify. (3 i)3
 0.74: Simplify. (2 i)3
 0.75: Simplify. (1 i)3
 0.76: Simplify. (4 3i)3
 0.77: Electrical impedance is the ratio of voltage to current in ac circu...
 0.78: Electrical impedance is the ratio of voltage to current in ac circu...
 0.79: Explain the mistake that is made.Write the quotient in standard for...
 0.80: Explain the mistake that is made. Write the product in standard for...
 0.81: Determine whether each statement is true or false.The product is a ...
 0.82: Determine whether each statement is true or false. Imaginary number...
 0.83: Determine whether each statement is true or false. Real numbers are...
 0.84: Determine whether each statement is true or false. There is no comp...
 0.85: Factor completely over the complex numbers: x4 2x2 1.
 0.86: Factor completely over the complex numbers: x4 18x2 81.
 0.87: Use a graphing utility to simplify the expression. Write your answe...
 0.88: Use a graphing utility to simplify the expression. Write your answe...
 0.89: Use a graphing utility to simplify the expression. Write your answe...
 0.90: Use a graphing utility to simplify the expression. Write your answe...
 0.1: Approximate to two decimal places by (a) rounding and (b) truncatin...
 0.2: Approximate to two decimal places by (a) rounding and (b) truncatin...
 0.3: Simplify. 7 2 5 4 3 5
 0.4: Simplify. 2(5 3) 7(3 2 5)
 0.5: Simplify. 16 (2)(4
 0.6: Simplify. 3(x y) 4(3x 2y)
 0.7: Perform the indicated operation and simplify.x 4 x 3
 0.8: Perform the indicated operation and simplify.y 3 +y 5 y 6
 0.9: Perform the indicated operation and simplify.12 7#21
 0.10: Perform the indicated operation and simplify.a2 b3 ,2a b2
 0.11: Simplify using properties of exponents. (2z)3
 0.12: Simplify using properties of exponents. (4z2)3
 0.13: Simplify using properties of exponents. (2x2 y3)2 (4xy)3
 0.14: Simplify using properties of exponents. (3x3 y2)2 2(x2y)
 0.15: Simplify using properties of exponents. Express 0.00000215 in scien...
 0.16: Simplify using properties of exponents. Express 7.2 109 as a real n...
 0.17: Perform the indicated operation and write the results in standard f...
 0.18: Perform the indicated operation and write the results in standard f...
 0.19: Perform the indicated operation and write the results in standard f...
 0.20: Perform the indicated operation and write the results in standard f...
 0.21: Perform the indicated operation and write the results in standard f...
 0.22: Perform the indicated operation and write the results in standard f...
 0.23: Perform the indicated operation and write the results in standard f...
 0.24: Perform the indicated operation and write the results in standard f...
 0.25: Perform the indicated operation and write the results in standard f...
 0.26: Perform the indicated operation and write the results in standard f...
 0.27: Perform the indicated operation and write the results in standard f...
 0.28: Perform the indicated operation and write the results in standard f...
 0.29: Factor out the common factor. 14x2y2 10xy3
 0.30: Factor out the common factor. 30x4 20x3 10x2
 0.31: Factor the trinomial into a product of two binomials. 2x2 9x 5
 0.32: Factor the trinomial into a product of two binomials. 6x2 19x 7
 0.33: Factor the trinomial into a product of two binomials. 16x2 25
 0.34: Factor the trinomial into a product of two binomials. 9x2 30x 25
 0.35: Factor the sum or difference of two cubes. x3 125
 0.36: Factor the sum or difference of two cubes. 1 8x3
 0.37: Factor into a product of three polynomials. 2x3 4x2 30x
 0.38: Factor into a product of three polynomials. 6x3 5x2 x
 0.39: Factor into a product of two binomials by grouping. x3 x2 2x 2
 0.40: Factor into a product of two binomials by grouping. 2x3 x2 6x 3
 0.41: State the domain restrictions on each of the rational expressions.4...
 0.42: State the domain restrictions on each of the rational expressions.1...
 0.43: Simplify.x2  4 x  2
 0.44: Simplify.x  5 x  5
 0.45: Simplify.t2 + t  6 t2  t 
 0.46: Simplify.z3  z z2 + z
 0.47: Perform the indicated operation and simplify.2 + 3x  10 x2 + 2x  ...
 0.48: Perform the indicated operation and simplify.x2  x  2 x3 + 3x2,x ...
 0.49: Perform the indicated operation and simplify.1 x + 11 x + 3
 0.50: Perform the indicated operation and simplify.1 x 1 x + 1+1 x + 2
 0.51: Perform the indicated operation and simplify.2 +1 x  31 5x  15+ 4
 0.52: Perform the indicated operation and simplify.1 x +2 x23 1 x2
 0.53: Simplify.
 0.54: Simplify.
 0.55: Simplify.
 0.56: Simplify.
 0.57: Simplify.
 0.58: Simplify.
 0.59: Simplify.
 0.60: Simplify.
 0.61: Simplify.
 0.62: Simplify.
 0.63: Simplify.
 0.64: Simplify.
 0.65: Simplify.
 0.66: Simplify.
 0.67: Simplify.
 0.68: Simplify.
 0.69: Simplify. i19
 0.70: Simplify. i9
 0.71: Perform the indicated operation, simplify, and express in standard ...
 0.72: Perform the indicated operation, simplify, and express in standard ...
 0.73: Perform the indicated operation, simplify, and express in standard ...
 0.74: Perform the indicated operation, simplify, and express in standard ...
 0.75: Perform the indicated operation, simplify, and express in standard ...
 0.76: Perform the indicated operation, simplify, and express in standard ...
 0.77: Perform the indicated operation, simplify, and express in standard ...
 0.78: Perform the indicated operation, simplify, and express in standard ...
 0.79: Express the quotient in standard form.1 2  i
 0.80: Express the quotient in standard form.1 3 + i
 0.81: Express the quotient in standard form.7 + 2i 4 + 5i
 0.82: Express the quotient in standard form.6  5i 3  2i
 0.83: Express the quotient in standard form.10 3i
 0.84: Express the quotient in standard form.7 2i
 0.85: Use your calculator to evaluate . Does the answer appear to be a ra...
 0.86: Use your calculator to evaluate . Does the answerappear to be a rat...
 0.87: Use a graphing utility to evaluate the expression. Express your ans...
 0.88: Use a graphing utility to evaluate the expression. Express your ans...
 0.89: Use a graphing utility to plot the graphs of the three expressions ...
 0.90: Use a graphing utility to plot the graphs of the three expressions ...
 0.91: Use a graphing utility to plot the graphs of the three expressions ...
 0.92: Use a graphing utility to plot the graphs of the three expressions ...
 0.93: For each given expression: (a) simplify the expression, (b) use a g...
 0.94: For each given expression: (a) simplify the expression, (b) use a g...
 0.95: Given a. Rationalize the denominator. b. Use a graphing utility to ...
 0.96: Given a. Rationalize the denominator. b. Use a graphing utility to ...
 0.97: Use a graphing utility to simplify the expression. Write your answe...
 0.98: Use a graphing utility to simplify the expression. Write your answe...
 0.99: Apply a graphing utility to simplify the expression and write your ...
 0.1: Simplify. 116
 0.2: Simplify. 2 3 54x6
 0.3: Simplify. 3(2 + 52) + 2(3  7)  (32  1)
 0.4: Simplify. 2 5 32
 0.5: Simplify. 212x2
 0.6: Simplify. i17
 0.7: Simplify. (x2 y3 z1)2 (x1 y2 z3)12
 0.8: Simplify. 31x  41x + 51x
 0.9: Simplify. 3118  4132
 0.10: Simplify. A516  212BA16 + 312B
 0.11: Perform the indicated operation and simplify. (3y2  5y + 7)  (y2 ...
 0.12: Perform the indicated operation and simplify. (2x 3)(5x 7)
 0.13: Factor. x2 16
 0.14: Factor. 3x2 15x 18
 0.15: Factor. 4x2 12xy 9y2
 0.16: Factor. x4 2x2 1
 0.17: Factor. 2x2 x 1
 0.18: Factor. 6y2 y 1
 0.19: Factor. 2t3 t2 3t
 0.20: Factor. 2x3 5x2 3x
 0.21: Factor. x2 3yx 4yx 12y2
 0.22: Factor. x4 5x2 3x2 15
 0.23: Factor. 81 3x3
 0.24: Factor. 27x x4
 0.25: Perform the indicated operations and simplify.2 x +3 x  1
 0.26: Perform the indicated operations and simplify.5x x2  7x + 104 x2  25
 0.27: Perform the indicated operations and simplify.x  1 x2  1 # x2 + x...
 0.28: Perform the indicated operations and simplify.4x2  9 x2  11x  60...
 0.29: Perform the indicated operations and simplify.x  3 2x  5,x2  9 5...
 0.30: Perform the indicated operations and simplify.1  t 3t + 1,t2  2t ...
 0.31: Write the resulting expression in standard form. (1 3i)(7 5i)
 0.32: Write the resulting expression in standard form. 2  11i 4 + i
 0.33: Write the resulting expression in standard form. Rationalize the de...
 0.34: Write the resulting expression in standard form. Represent 0.000015...
 0.35: Write the resulting expression in standard form. Simplify and state...
 0.36: Write the resulting expression in standard form. For the given expr...
 0.37: Write the resulting expression in standard form. Apply a graphing u...
Solutions for Chapter 0: Prerequisites and Review
Full solutions for Algebra and Trigonometry  3rd Edition
ISBN: 9780470648032
Solutions for Chapter 0: Prerequisites and Review
Get Full SolutionsChapter 0: Prerequisites and Review includes 802 full stepbystep solutions. Since 802 problems in chapter 0: Prerequisites and Review have been answered, more than 59811 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9780470648032.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.