Number Comparison: Identifying the Largest Value Among Given Options

Determine the largest value from among the given options:

In order to determine which of the suggested options has the largest numerical value, we will apply the following root law:

$\sqrt[\textcolor{blue}{n}]{a^{\textcolor{red}{m}}}=a^{\frac{\textcolor{red}{m}}{\textcolor{blue}{n}}} =(\sqrt[\textcolor{blue}{n}]{a})^{\textcolor{red}{m}}$

Let's start by applying this law to each of the suggested options (and remember that a square root is a second-order root - which we don't explicitly write next to the root), meaning - we will convert the roots to exponential notation, then we'll use the (known) root of the number 25:

$\sqrt{25^{\textcolor{red}{2}}}=25^{\frac{\textcolor{red}{2}}{\textcolor{blue}{2}}}=(\sqrt{25})^{\textcolor{red}{2}}=5^2 \\ \sqrt{25}=\sqrt{25^{\textcolor{red}{1}}}=25^{\frac{\textcolor{red}{1}}{\textcolor{blue}{2}}}=(\sqrt{25})^{\textcolor{red}{1}}=5^1 \\ \sqrt{25^{\textcolor{red}{3}}}=25^{\frac{\textcolor{red}{3}}{\textcolor{blue}{2}}}=(\sqrt{25})^{\textcolor{red}{3}}=5^3 \\ \sqrt{25^{\textcolor{red}{4}}}=25^{\frac{\textcolor{red}{4}}{\textcolor{blue}{2}}}=(\sqrt{25})^{\textcolor{red}{4}}=5^4 \\$We obtained four options which are all powers of the same base (5), since this base is greater than 1, the largest option will be the one where the base (5) is raised to the highest power (and the opposite if the base is between 0 and 1, then as the power increases the value of the number - meaning the base raised to that power - decreases),

Therefore:

$5^4>5^3>5^2>5^1$

Thus answer D is the correct answer.