2016 AMC 10A Problems/Problem 2

Problem

For what value of $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$ ?

$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Solution

We can rewrite $10^{x}\cdot 100^{2x}=1000^{5}$ as $10^{5x}=10^{15}$ : $10^x\cdot100^{2x}=10^x\cdot(10^2)^{2x}$ $10^x\cdot10^{4x}=(10^3)^5$ $10^{5x}=10^{15}$ Since the bases are equal, we can set the exponents equal: $5x=15$ . Solving gives us: $x = \boxed{\textbf{(C)}\;3.}$

2016 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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2016 AMC 10A Problems/Problem 2

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