Question

question
select the expression that is equivalent to (sqrt5{2} cdot sqrt5{2^2}).

answer
(\bigcirc) (2^{\frac{2}{25}}) (\bigcirc) (2^{\frac{3}{5}})
(\bigcirc) (4^{\frac{2}{25}}) (\bigcirc) (4^{\frac{3}{5}})

Explanation:

Step1: Convert roots to exponents

Recall that $\sqrt[n]{a}=a^{\frac{1}{n}}$, so:
$\sqrt[5]{2}=2^{\frac{1}{5}}$, $\sqrt[5]{2^2}=(2^2)^{\frac{1}{5}}=2^{\frac{2}{5}}$

Step2: Multiply exponential terms

When multiplying like bases, add exponents:
$2^{\frac{1}{5}} \cdot 2^{\frac{2}{5}} = 2^{\frac{1}{5}+\frac{2}{5}}$

Step3: Simplify the exponent

$\frac{1}{5}+\frac{2}{5}=\frac{3}{5}$, so the expression becomes $2^{\frac{3}{5}}$

Answer:

$2^{\frac{3}{5}}$ (the second option)