# Lesson Explainer: Square Roots of Perfect Squares Mathematics • 8th Grade

In this explainer, we will learn how to find square roots of perfect square integers, fractions, and decimals.

To understand square roots, we need to recall what squaring a number is.

### Definition: Squaring a Number

Squaring a number consists in multiplying this number by itself. As the name suggests, a number squared is the area of a square with this number for a side.

For example, three squared (written ) is the area of the square of side three.

### Definition: Square Root of a Number

Taking the square root is the inverse operation of squaring a number. It means that taking the square root of a given number is finding the side of the square whose area is this number.

For instance, taking the square root of twenty-five (written ) is finding the length of the side of the square whose area is 25. We find it is 5, as illustrated in the diagram below.

Note that, in this lesson, we will only deal with square roots that are integers or simple fractions (that is, fractions whose numerator and denominator are integers).

Let us work out a first simple question.

### Example 1: Taking the Square Root of a Square Number

Calculate the following: .

### Answer

We are considering here a square of area 4. We see that a square of side 2 does have an area of . So .

We are going to look now at a more complex example that will allow us to introduce an important calculation rule.

Calculate .

### Answer

In this question, we want to find the side of a square whose area is .

Let us start with a square of area 81. It is a square of side (check: ).

Taking 16 of such squares gives this big square:

This big square has an area of , and we find that the side length of this square is .

Let us look back at how we were able to draw this square. We found the side of the red square by taking the square root of 81. And there are red squares per row and per column in the big square to have 16 red squares all together.

We can conclude that

The result we have found in the previous example can be generalized to all real numbers.

### How To: Finding the Square Root of a Product

Let us look now at an example with a simple fraction.

Calculate .

### Answer

Let us draw a square of area . We start with a square of area 196. Its side is (check: ).

We need to divide this square into 25 to get a square of area . For this, we split each side in parts. We get indeed 25 small squares, one of which is shaded in green in the diagram below.

The small square (shaded in green) has an area of and its side is one-fifth of the side of the big (red) square, that is, .

We have found that .

The result we have found in the previous example can be generalized to all real numbers.

### How To: Finding the Square Root of a Quotient

In the next example, we are going to see how to deal with square roots of decimal numbers.

Calculate .

### Answer

We want to find the opposite (that is, with an opposite sign) of the square root of 0.25. It is very useful here to write 0.25 as a fraction:

We can then write

And so

As we are asked to find , our answer is

Note that we have as well .

Also, using decimal numbers instead of fractions gives, of course, the same result: , so .

Calculate .

### Answer

To calculate this square root, let us first write as an improper fraction:

Now, we can apply the quotient rule:

The answer is .

Let us summarize this explainer.

### Key Points

• Squaring a number consists in multiplying this number by itself. A number squared is the area of a square with this number for a side.
• Taking the square root is the inverse operation of squaring a number. It means that taking the square root of a given number is finding the side of the square whose area is this number.
• Product rule: .
• Quotient rule: .
• To find the square root of a decimal without a calculator, it is helpful to write this decimal as a fraction and then apply the quotient rule.
• To find the square root of a mixed number, write first the mixed number as an improper fraction and then apply the quotient rule.

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