Know Your Powers
Given these equations with unknown powers $x$ and $y$, can you work out $xy$?
Age
14 to 16
Challenge level
Problem
Given that $4^x=9$ and $9^y=64$, what is the value of $xy$?
This problem is taken from the UKMT Mathematical Challenges.
Student Solutions
Answer: $xy=3$
$4^x=9$ can be substituted into $9^y=64$, giving $\left(4^x\right)^y=64$.
Finding a power of a power of a number is the same as multiplying the powers, so $\left(4^x\right)^y=4^{xy}$. So $4^{xy}=64$.
$4^1=4$
$4^2=16$
$4^3=16\times4=64$
So $xy=3$.