College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 6 - Section 6.3 - Exponential Functions - 6.3 Assess Your Understanding - Page 436: 88



Work Step by Step

Note that $-3x=x(-3)$. Thus, $2^{-3x}$ can be written as $2^{x(-3)}$. RECALL: (1) $a^{mn} = (a^m)^n$ (2) $a^m = b^m \longrightarrow a=b$ (3) $\dfrac{1}{a^m} = a^{-m}$ Use rule (1) above to obtain: $2^{x(-3)} = (2^x)^{-3}$ Thus, $2^{-3x}$ in the given equation may be replaced with its equivalent $(2^x)^{-3}$ to obtain: $(2^x)^{-3} = \dfrac{1}{1000}$ Note that $1000=10^3$, so the expression above is equivalent to: $(2^x)^{-3} = \dfrac{1}{10^3}$ Use rule (3) above to obtain: $(2^x)^{-3} = 10^{-3}$ Use rule (2) above to obtain: $2^x=10$
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