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 Pages**

Rationalize The Denominator

Rationalising Surds

**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.

**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:**

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

**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:**

**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.

Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.