1Cademy - Simplifying $$\sqrt{\frac{48m^7n^2}{100m^5n^8}}$$

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Procedure for Simplifying the Quotient of Radical Expressions

Example

Simplifying $\sqrt{\frac{48m^7n^2}{100m^5n^8}}$

To simplify a square root whose radicand is a fraction containing numerical coefficients and multiple variables, first reduce the fraction under the radical, then apply the Quotient Property of Radical Expressions.

$\sqrt{\frac{48m^7n^2}{100m^5n^8}}$

Step 1 — Simplify the fraction in the radicand. Divide out common numerical factors. The coefficients $48$ and $100$ share a common factor of $4$ , leaving $\frac{12}{25}$ . Apply the Quotient Property for Exponents to the variables: $\frac{m^7}{m^5} = m^2$ in the numerator, and $\frac{n^2}{n^8} = \frac{1}{n^6}$ in the denominator. The expression becomes: $\sqrt{\frac{12m^2}{25n^6}}$

Step 2 — Rewrite using the Quotient Property. Split the single radical into a quotient of two separate radicals: $\frac{\sqrt{12m^2}}{\sqrt{25n^6}}$

Step 3 — Simplify the radicals in the numerator and the denominator. The denominator is a perfect square: $\sqrt{25n^6} = 5|n^3|$ . For the numerator, factor $12m^2$ into a perfect square and a remaining factor: $\sqrt{4m^2 \cdot 3}$ . Separate the radicals: $\sqrt{4m^2} \cdot \sqrt{3}$ . Evaluate the perfect square: $2|m|\sqrt{3}$ .

Step 4 — Simplify the final expression. $\frac{2|m|\sqrt{3}}{5|n^3|}$

The simplified result is $\frac{2|m|\sqrt{3}}{5|n^3|}$ . Reducing the fraction under the radical first simplifies the variables and reveals a perfect square in the denominator.

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Updated 2026-05-25

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OpenStax Intermediate Algebra

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