1Cademy - Simplifying $$\sqrt{\frac{11}{28}}$$

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Rationalizing the Denominator

Example

Simplifying $\sqrt{\frac{11}{28}}$

Simplify a square root whose radicand is a fraction that is not a perfect square, by applying the Quotient Property, simplifying the denominator, and then rationalizing.

$\sqrt{\frac{11}{28}}$

Step 1 — Rewrite using the Quotient Property. Since $\frac{11}{28}$ is not a perfect square fraction, separate the radical into a quotient of two square roots:

$\sqrt{\frac{11}{28}} = \frac{\sqrt{11}}{\sqrt{28}}$

Step 2 — Simplify the denominator. The largest perfect square factor of $28$ is $4$ : $\sqrt{28} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7}$ . The expression becomes:

$\frac{\sqrt{11}}{2\sqrt{7}}$

Step 3 — Rationalize the denominator. Multiply both the numerator and denominator by $\sqrt{7}$ :

$\frac{\sqrt{11} \cdot \sqrt{7}}{2\sqrt{7} \cdot \sqrt{7}} = \frac{\sqrt{77}}{2 \cdot 7}$

Step 4 — Simplify. Multiply in the denominator:

$\frac{\sqrt{77}}{14}$

The result is $\frac{\sqrt{77}}{14}$ . This example follows the same pattern as simplifying $\sqrt{\frac{5}{12}}$ : apply the Quotient Property, simplify the denominator radical by extracting perfect square factors, and then rationalize the remaining radical in the denominator. The product $\sqrt{11} \cdot \sqrt{7} = \sqrt{77}$ stays under the radical because $77$ has no perfect square factors.

Updated 2026-04-21

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OpenStax Elementary Algebra

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