1Cademy - Product Property of $$n$$th Roots

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Product Property of Square Roots

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Product Property of $n$ th Roots

The Product Property of $n$ th Roots extends the Product Property of Square Roots to radicals with any index $n \geq 2$ . When $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers and $n$ is an integer with $n \geq 2$ :

$\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \qquad \text{and} \qquad \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$

Read left to right, the property splits the $n$ th root of a product into a product of two $n$ th roots. Read right to left, it combines two $n$ th roots with the same index into a single radical.

This property is the primary tool for simplifying higher roots: it allows any perfect $n$ th power factor to be separated from the radicand and evaluated independently. The simplification procedure mirrors the three-step process used for square roots — rewrite the radicand as a product using the largest perfect $n$ th power factor, split into two radicals using this property, then simplify the radical containing the perfect $n$ th power.

Updated 2026-05-01

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