Simplify Squares Roots (Radicals) That Have Fractions

How to find the square root of a fraction?

Key Rule: The Quotient Property of Square Roots
\(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\)

This rule states that the square root of a fraction is a fraction of square roots. It allows you to separate the square root of a fraction into the square root of the numerator divided by the square root of the denominator. This is the first step in simplifying.

The following are some of the steps that can be used to find the square root of a fraction:

Separate Numerator & Denominator

A. Both numerator and denominator are perfect squares. Find the square root of the numerator and denominator separately.

Example:
\(\sqrt {\frac{{16}}{{25}}} = \frac{{\sqrt {16} }}{{\sqrt {25} }} = \frac{{\sqrt {{4^2}} }}{{\sqrt {{5^2}} }} = \frac{4}{5}\)

Reduce Fraction

B. Some fractions can be reduced to fractions with perfect squares as the numerator and denominator. Then, the square root of the simplified fraction can be determined as shown in the example below.

Example:
\(\sqrt {\frac{{18}}{{50}}} = \sqrt {\frac{9}{{25}}} = \frac{{\sqrt {{3^2}} }}{{\sqrt {{5^2}} }} = \frac{3}{5}\)

Change to Improper Fraction

C. For a mixed number, change it to an improper fraction before calculating the square root.

Example:
\(\sqrt {1\frac{{13}}{{36}}} = \sqrt {\frac{{49}}{{36}}} = \frac{{\sqrt {{7^2}} }}{{\sqrt {{6^2}} }} = \frac{7}{6} = 1\frac{1}{6}\)

Example:
Calculate the value of each of the following:
\(\begin{array}{l}{\rm{a)}}\,\,\sqrt {\frac{{25}}{{36}}} \\{\rm{b)}}\,\,\sqrt {\frac{{18}}{{32}}} \\{\rm{c)}}\,\,\sqrt {1\frac{{11}}{{25}}} \end{array}\)

Solution:

Simplify Square Root Worksheets
Simplify square roots
Rationalize denominators

How to simplify fractions inside a square root?

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How to simplify square roots in the denominator of a fraction?

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Square Roots of Fractions/Rational Numbers

Finding the square of rational numbers for perfect squares as well as estimating non-perfect squares.

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Quotient Rule & Simplifying Square Roots

An introduction to the quotient rule for square roots and radicals and how to use it to simplify expressions containing radicals.

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