The nth Root and Fractional Indices

The cube root of 125 is 5, because $5^3=125$. Therefore, $125^{\frac{1}{3}}=5$.

The cube root of 64 is 4, because $4^3=64$.

The cube root of 27 is 3, because $3^3=27$. Then, raising 3 to the power of 2 gives 9.

First, find $\sqrt[4]{16}$, which is 2 because $2^4=16$. Then, raise 2 to the power of 3: $2^3=8$. So, $16^{\frac{3}{4}}=8$.

$125^{\frac{2}{3}}$ means first taking the cube root of 125, which is 5, and then squaring the result. $5^2=25$. So, $125^{\frac{2}{3}}=25$.

The fourth root of 16 is 2, and raising 2 to the power of 5 gives 32. So, $16^{\frac{5}{4}}=32$.