Let $2^x$ be the greatest power of $2$ that is a factor of $144$, and let $3^y$ be the greatest power of $3$ that is a factor of $144$. Evaluate the following expression: $$\left(\frac15\right)^{y - x}$$
To find $2^x$ and $3^y$, I need to factorize $144$ into prime factors. I know that $144 = 12 \times 12$, and that $12 = 2 \times 2 \times 3$. So $144 = 2^4 \times 3^2$. Therefore, $x = 4$ and $y = 2$. Now I can plug these values into the expression: $$\left(\frac15\right)^{y - x} = \left(\frac15\right)^{2 - 4}$$ This simplifies to: $$\left(\frac15\right)^{-2} = \frac{1}{(1/5)^2} = \frac{1}{(1/25)} = 25$$ 