1Cademy - Simplifying $$\sqrt{25y^5}$$

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Simplifying $\sqrt{x^3}$

Example

Simplifying $\sqrt{25y^5}$

Simplify a square root whose radicand contains a numerical perfect square multiplied by a variable raised to an odd power, by applying the Product Property of Square Roots.

$\sqrt{25y^5}$

Step 1 — Find the largest perfect square factor. The coefficient $25$ is already a perfect square ( $5^2 = 25$ ). For the variable $y^5$ , the largest even power less than $5$ is $y^4$ , so the largest perfect square factor of the radicand is $25y^4$ . Rewrite:

$\sqrt{25y^5} = \sqrt{25y^4 \cdot y}$

Step 2 — Apply the Product Property. Split the radical into a product of two radicals:

$\sqrt{25y^4 \cdot y} = \sqrt{25y^4} \cdot \sqrt{y}$

Step 3 — Simplify. Since $\sqrt{25} = 5$ and $\sqrt{y^4} = y^2$ :

$\sqrt{25y^4} \cdot \sqrt{y} = 5y^2\sqrt{y}$

The simplified form is $5y^2\sqrt{y}$ . This example extends the odd-exponent technique from $\sqrt{x^3}$ to a radicand that also includes a perfect square coefficient — both the numerical and variable parts are simplified independently using the same three-step procedure.

Updated 2026-04-21

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OpenStax Elementary Algebra

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