Simplify Squares Roots (Radicals) That Have Fractions
Related Pages
Printable & Online Square Roots & Radicals Worksheets
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.
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?
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 out our new and fun Fraction Concoction Game.
Add and subtract fractions to make exciting fraction concoctions following a recipe. There are four levels of difficulty: Easy, medium, hard and insane. Practice the basics of fraction addition and subtraction or challenge yourself with the insane level.
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