2016 AMC 10A Problem 2

The numbers $10$, $100$ and $1000$ are all powers of $10$, so rewrite everything with base $10$.

1. $100 = 10^2$, so $100^{2x} = (10^2)^{2x}$. A power of a power multiplies the exponents, so this is $10^{4x}$.
2. $1000 = 10^3$, so $1000^5 = (10^3)^5 = 10^{15}$.
3. The left side is now $10^x \cdot 10^{4x}$. Multiplying powers of the same base adds the exponents, so it is $10^{5x}$.
4. The equation reads $10^{5x} = 10^{15}$. Two powers of $10$ are equal only when their exponents are equal, so $5x = 15$ and $x = 3$.

Check: $10^3 \cdot 100^6 = 10^3 \cdot 10^{12} = 10^{15}$, and $1000^5 = 10^{15}$ too.

The answer is $\boxed{\textbf{(C)}\ 3}$.

When every number in the equation is a power of the same base, rewrite them all with that base; then you only need to compare exponents. Keep the two rules apart: a power raised to a power multiplies the exponents, and two powers multiplied together add them.

Try the answer choices. With $x = 3$ the left side is $10^3 \cdot 100^6 = 1000 \cdot 10^{12} = 10^{15}$, which equals $1000^5$. It works on the first try.

Express everything with base $10$: $100 = 10^2$ and $1000 = 10^3$, so
$$100^{2x} = (10^2)^{2x} = 10^{4x}, \qquad 1000^5 = (10^3)^5 = 10^{15}.$$

The left side becomes $10^x \cdot 10^{4x} = 10^{5x}$. Setting $10^{5x} = 10^{15}$ gives $5x = 15$, so $x = 3$.

The answer is $\boxed{\textbf{(C)}\ 3}$.