# Simplify square roots (radicals) that have fractions

In these lessons, we will look at some examples of simplifying fractions within a square root (or radical). Some techniques used are: find the square root of the numerator and denominator separately, reduce the fraction and change to improper fraction.

Related Topics:
More Lessons on Fractions
Fraction Worksheets
Fraction Games

### Separate Numerator & Denominator

A. The square root of some fractions can be determined by finding the square root of the numerator and denominator separately.

Example:

$$\sqrt {\frac{{16}}{{25}}} = \frac{{\sqrt {16} }}{{\sqrt {25} }} = \frac{{\sqrt {{4^2}} }}{{\sqrt {{5^2}} }} = \frac{4}{5}$$

For any positive number x and y,

$$\sqrt {\frac{x}{y}} = \frac{{\sqrt x }}{{\sqrt y }}$$

In other words, the square root of a fraction is a fraction of square roots.

### 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 finding 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:

a) Work out the square roots of the numerator and denominator separately.
$$\begin{array}{c}\sqrt {\frac{{25}}{{36}}} = \frac{{\sqrt {25} }}{{\sqrt {36} }}\\ = \frac{{\sqrt {{5^2}} }}{{\sqrt {{6^2}} }}\\ = \frac{5}{6}\end{array}$$

b) Reduce the fraction first.
$$\begin{array}{c}\sqrt {\frac{{18}}{{32}}} = \sqrt {\frac{9}{{16}}} \\ = \frac{{\sqrt 9 }}{{\sqrt {16} }}\\ = \frac{{\sqrt {{3^2}} }}{{\sqrt {{4^2}} }}\\ = \frac{3}{4}\end{array}$$

c) Change to improper fraction first.
$$\begin{array}{c}\sqrt {1\frac{{11}}{{25}}} = \sqrt {\frac{{36}}{{25}}} \\ = \frac{{\sqrt {36} }}{{\sqrt {25} }}\\ = \frac{{\sqrt {{6^2}} }}{{\sqrt {{5^2}} }}\\ = \frac{6}{5}\\ = 1\frac{1}{5}\end{array}$$

How to simplify fractions inside a square root? How to simplify square roots in the denominator of a fraction? Square Roots of Fractions/Rational Numbers
Finding the square of rational numbers for perfect squares as well as estimating non-perfect squares. 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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