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# X y 2 differences and sum of cubes x y x 2 xy y 2 x 3

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x - y ) 2 Differences and Sum of Cubes ( x + y )( x 2 - xy + y 2 ) = x 3 + y 3 ( x - y )( x 2 + xy + y 2 ) = x 3 - y 3 MBA501: Math and Stat for Business 41 / 71

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§ 1.4 Rational Expressions Rational Expressions Expression that can be written as the quotient of two polynomials x + 1 x 2 + 3 MBA501: Math and Stat for Business 42 / 71
§ 1.4 Rational Expressions Rational Expressions Expression that can be written as the quotient of two polynomials x + 1 x 2 + 3 Operations of Rational Expressions Cancellation Property PS QS = P Q Multiplication Rule P Q · R S = PR QS Division Rule P Q ÷ R S = P Q · S R MBA501: Math and Stat for Business 43 / 71

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§ 1.4 Rational Expressions Multiply, divide, add or subtract 21) 15 p - 3 6 ÷ 10 p - 2 3 27) k 2 - k - 6 k 2 + k - 12 · k 2 + 3 k - 4 k 2 + 2 k - 3 39) 7 ( b + 2 ) + 2 5 ( b + 2 ) MBA501: Math and Stat for Business 44 / 71
§ 1.5 Exponents and Radicals What We Know a n = a · a · a · · · a n real number a is the base natural number n is the exponent Extension Expand the definition of exponent to include negative exponents and rational-number exponents ( - 2 ) - 2 , 8 1 / 3 MBA501: Math and Stat for Business 45 / 71

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§ 1.5 Exponents and Radicals Negative Exponent If n is a natural number, and if a = 0, then: a - n = 1 a n . Inversion Property ( a b ) - n = ( b a ) n . MBA501: Math and Stat for Business 46 / 71
§ 1.5 Exponents and Radicals Exercises ( 1 3 ) - 2 ( rs ) 3 r 4 2 MBA501: Math and Stat for Business 47 / 71

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§ 1.5 Exponents and Radicals Roots Definition If a is a real number and n is a positive integer, then a 1 / n is defined to be the n th root of a (if it exists). Note if a is negative and n is even, then n th root of a does NOT exist. If n is even, then n th root of a is the positive real number whose n th power is a 0 . 4 = 16 = ( 16 ) 1 / 2 ⇐⇒ 4 2 = 16 If n is odd, then n th root of a is the real number whose n th power is a . 2 = ( 8 ) 1 / 3 , - 2 = ( - 8 ) 1 / 3 MBA501: Math and Stat for Business 48 / 71
§ 1.5 Exponents and Radicals Radicals If any real number a and natural number n , is an even and a 0, or if n is an odd natural number, a 1 / n = n a . For all rational numbers m / n and all real numbers a for which n a exists, a m / n = ( n a ) m , or a m / n = ( n a m ) MBA501: Math and Stat for Business 49 / 71

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§ 1.6 First-Degree Equations Equations An equation is a statement that two mathematical expressions are equal 2 x - 1 = 3 x - 2 MBA501: Math and Stat for Business 50 / 71
§ 1.6 First-Degree Equations Equations An equation is a statement that two mathematical expressions are equal 2 x - 1 = 3 x - 2 First-Degree Equations An equation involve only constant and first power of the variable. 2 x + 1 = 3 MBA501: Math and Stat for Business 51 / 71

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§ 1.6 First-Degree Equations Equations An equation is a statement that two mathematical expressions are equal 2 x - 1 = 3 x - 2 First-Degree Equations An equation involve only constant and first power of the variable. 2 x + 1 = 3 Solution of an Equation An number that can be substituted for the variable to make the equation true.
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