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Simplify Square Roots (radicals) that have Fractions



In this lesson, 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 :

For any positive number x and y,

In other words, the square root of a fraction is a fraction of square roots. This is also called the Quotient Rule.

The following video shows how to simplify fractions inside a square root by splitting it into two square roots.

The converse is also true and can be used to simplify square roots.
Quotient Rule & Simplifying Square Roots
This video gives an introduction to the quotient rule for square roots and radicals and how to use it to simplify expressions containing radicals.

The following video shows how to simplify square roots in the denominator of a fraction (also called rationalizing the denominator).


 



Reduce Fraction

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

Example :

square root fraction

Change to Improper Fraction

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

Example :

Example:

Calculate the value of the following:

Solution:

Square Roots of Fractions/Rational Numbers
The following video shows how to find the square of rational numbers for perfect squares as well as estimating non-perfect squares.


 



Square Root of a Quotient or Fraction

This video will show how to simplify square roots of fractions by multiplying to get a perfect square in the denominator in order to eliminate the radical there.

This video gives more examples of simplifying square root of fractions by using the Quotient Property of Square Roots

This video gives more examples of how to rationalize a denominator


 



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