1Cademy - Simplifying $$\sqrt{72n^7}$$

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Simplifying $\sqrt{x^3}$

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Simplifying $\sqrt{72n^7}$

Simplify a square root whose radicand has a non-perfect-square coefficient and a variable raised to an odd power, so that both the constant and the variable require factoring to extract perfect square factors.

$\sqrt{72n^7}$

Step 1 — Find the largest perfect square factor. The largest perfect square that divides $72$ is $36$ (since $6^2 = 36$ and $72 = 36 \cdot 2$ ). For $n^7$ , the largest even power is $n^6$ . Together, the largest perfect square factor of the radicand is $36n^6$ :

$\sqrt{72n^7} = \sqrt{36n^6 \cdot 2n}$

Step 2 — Apply the Product Property. Separate the radical:

$\sqrt{36n^6 \cdot 2n} = \sqrt{36n^6} \cdot \sqrt{2n}$

Step 3 — Simplify. Since $\sqrt{36} = 6$ and $\sqrt{n^6} = n^3$ :

$\sqrt{36n^6} \cdot \sqrt{2n} = 6n^3\sqrt{2n}$

The simplified form is $6n^3\sqrt{2n}$ . Unlike the previous example where the coefficient was itself a perfect square, here the coefficient $72$ must be factored into its largest perfect square factor ( $36$ ) and a leftover factor ( $2$ ), which stays under the radical alongside the remaining variable factor.

Updated 2026-04-21

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OpenStax Elementary Algebra

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