College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter P - Prerequisites: Fundamental Concepts of Algebra - Exercise Set P.2: 111

Answer

$\frac{1}{x^{12}y^{16}z^{20}}$

Work Step by Step

$$(\frac{x^{3}y^{4}z^{5}}{x^{-3}y^{-4}z^{-5}})^{-2}$$ Within the parentheses, group like terms and simplify. $$=[(\frac{x^{3}}{x^{-3}})(\frac{y^{4}}{y^{-4}})(\frac{z^{5}}{z^{-5}})]^{-2}$$ When dividing exponential expressions with the same non-zero base, subtract the exponent of the denominator from the exponent of the numerator. $$=(x^{(3-(-3))}y^{(4-(-4))}z^{(5-(-5))})^{-2}$$ $$=(x^{(3+3)}y^{(4+4)}z^{(5+5)})^{-2}$$ $$=(x^{6}y^{8}z^{10})^{-2}$$ When a product is raised to an exponent, raise each factor to that exponent. $$=x^{(6\times(-2))}y^{(8\times(-2))}z^{(10\times(-2))}$$ $$=x^{-12}y^{-16}z^{-20}$$ When an exponent is negative, write the expression as a fraction and move the base from the numerator to the denominator (or vise versa) making the exponent positive. $$=\frac{1}{x^{12}y^{16}z^{20}}$$
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