Question

which expression is equivalent to $25^{\frac{2}{5}}$?\
options: $\sqrt5{25^2}$, $\frac{2}{\sqrt5{25}}$, $2\sqrt5{25}$, $\sqrt{25^5}$

Explanation:

Step1: Recall the formula for rational exponents

The formula for converting a rational exponent \(a^{\frac{m}{n}}\) to a radical is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) (where \(n\) is the index of the radical and \(m\) is the exponent of the base inside the radical, for \(a\geq0\), \(n>0\)).

Step2: Apply the formula to \(25^{\frac{2}{5}}\)

Here, \(a = 25\), \(m = 2\), and \(n=5\). Using the formula \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), we substitute the values: \(25^{\frac{2}{5}}=\sqrt[5]{25^{2}}\).

Answer:

\(\boldsymbol{\sqrt[5]{25^{2}}}\) (the first option among the given choices)