Question

1. which of the following is equivalent to $25^{\frac{7}{2}}$?

$\sqrt{25^{7}}$
$\sqrt7{25^{2}}$
$25 - \frac{2}{7}$
$25 - \frac{7}{2}$

Explanation:

Step1: Recall rational exponent rule

For any positive real number $a$, and integers $m,n$ where $n>0$, $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$.

Step2: Apply rule to given expression

For $25^{\frac{7}{2}}$, set $a=25$, $m=7$, $n=2$.
$25^{\frac{7}{2}} = \sqrt[2]{25^7} = \sqrt{25^7}$

Step3: Eliminate incorrect options

Subtraction options do not match exponent rules, and $\sqrt[7]{25^2}$ corresponds to $25^{\frac{2}{7}}$, which is not equivalent.

Answer:

$\boldsymbol{\sqrt{25^7}}$ (the first option)