Simplifying Square Roots When not a Perfect Square

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  • 0:05 Rules of Mathematics
  • 1:04 Perfect Square
  • 1:17 Imperfect Square
  • 1:39 Evaluating the Square…
  • 4:08 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Numbers that are imperfect squares are those that, when evaluated, do not give solutions that are integers. The proper mathematical way to simplify these imperfect squares is discussed in this lesson.

Rules of Mathematics

The study of mathematics is universal. It is studied the same whether you are in America, Russia, India or any other corner of the globe. To make it easy for all these mathematicians who speak different languages and come from different backgrounds, there are universal rules that apply to writing mathematical equations and the way in which all math is formatted. This way, mathematicians in Korea can understand the work that is being done in Canada.

One of these rules relates to how the square roots of imperfect squares are written. The proper mathematical way to write imperfect square roots is by simplifying them as much as possible without using a fraction or a decimal. This will result in numbers like the following:



Before we can work out how to simplify imperfect square roots, we need to do a bit of review about perfect and imperfect square roots.

Perfect Square

A perfect square is a number whose square root is an integer. An integer is a number that does not contain a fraction or a decimal. The table below shows the first 10 perfect squares.

Table showing first 10 perfect squares
table showing ten perfect squares

Imperfect Square

As you can see, only a few numbers are perfect squares. The rest can be classified as imperfect squares. Imperfect squares are numbers whose square roots contain fractions or decimals.

For example:

√20 = 4 ½

√77 = 8.77

Evaluating the Square Root of an Imperfect Square

A calculator will not simplify the square root of an imperfect square properly. It will give a result, but it will be in decimal form. So, to evaluate and simplify an imperfect square, you need to follow these steps:

1. Factor the number completely.

An easy way to factor a number is by using a factor tree. A factor tree can be created by writing down the number you want to factor and drawing two lines coming down from that number. Then, write two factors of that number under the lines. Continue on until only prime numbers remain. A prime number is one which cannot be reduced any smaller. The purpose of the factor tree is to determine which numbers can be removed from under the square root symbol.

2. Match up pairs of the same number.

Any numbers with a partner are perfect squares, and you can take the square root of these numbers.

3. Numbers without a partner remain under the square root.

These numbers cannot be simplified further.

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