1Cademy - Simplifying $$\sqrt{\frac{5}{12}}$$

Learn Before

Rationalizing the Denominator

Example

Simplifying $\sqrt{\frac{5}{12}}$

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.

$\sqrt{\frac{5}{12}}$

Step 1 — Rewrite using the Quotient Property. The fraction $\frac{5}{12}$ is not a perfect square, so split the radical into a quotient of two separate square roots:

$\sqrt{\frac{5}{12}} = \frac{\sqrt{5}}{\sqrt{12}}$

Step 2 — Simplify the denominator. The largest perfect square factor of $12$ is $4$ : $\sqrt{12} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$ . The expression becomes:

$\frac{\sqrt{5}}{2\sqrt{3}}$

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

$\frac{\sqrt{5} \cdot \sqrt{3}}{2\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{15}}{2 \cdot 3}$

Step 4 — Simplify. Multiply in the denominator:

$\frac{\sqrt{15}}{6}$

The result is $\frac{\sqrt{15}}{6}$ . When the fraction under the radical is not a perfect square and does not reduce to one, the Quotient Property separates the numerator and denominator into individual radicals. After simplifying the denominator by extracting any perfect square factors, the remaining radical in the denominator must be eliminated by rationalization.

Updated 2026-04-21

Contributors are:

Who are from:

References

OpenStax Elementary Algebra

Learn Before

Related

Learn After