Power up to 1000
Find $x+y$, where $2^x\times5^y=1000$
Problem
Student Solutions
Answer: $6$
Index laws
$\begin{align}1000&=10^3\\
&=\left(2\times5\right)^3\\
&=2^3\times5^3\end{align}$
So $x=3$ and $y=3$, so $x+y=6$.
Calculating directly
$\begin{align}1000&=10^3\\
&=10\times10\times10\\
&=\left(2\times5\right)\times\left(2\times5\right)\times\left(2\times5\right)\\
&=2\times2\times2\times5\times5\times5\\
&=2^3\times5^3\end{align}$
So $x=3$ and $y=3$, so $x+y=6$.