Explainer: Square Roots of Perfect Squares

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 3) 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 25) 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: 4.

Answer

We are considering here a square of area 4. We see that a square of side 2 does have an area of 22=4. So 4=2.

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

Example 2: Taking the Square Root of the Product of Two Square Numbers

Calculate 81×16.

Answer

In this question, we want to find the side of a square whose area is 81×16.

Let us start with a square of area 81. It is a square of side 81=9 (check: 9=99=81).

Taking 16 of such squares gives this big square:

This big square has an area of 81×16, and we find that the side length of this square is 9×4=36.

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 16=4 red squares per row and per column in the big square to have 16 red squares all together.

We can conclude that 81×16=81×16,81×16=9×4=36.

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

How To: Find the Square Root of a Product

𝑎×𝑏=𝑎×𝑏.

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

Example 3: Taking the Square Root of a Simple Fraction

Calculate 19625.

Answer

Let us draw a square of area 19625. We start with a square of area 196. Its side is 196=14 (check: 14=1414=196).

We need to divide this square into 25 to get a square of area 19625. For this, we split each side in 25=5 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 19625 and its side is one-fifth of the side of the big (red) square, that is, 145.

We have found that 19625=19625=145.

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

How To: Find the Square Root of a Quotient

𝑎𝑏=𝑎𝑏.

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

Example 4: Taking the Square Root of a Decimal

Calculate 0.25.

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: 0.25=14.

We can then write 0.25=14=14.

And so 0.25=12.

As we are asked to find 0.25, our answer is 0.25=12.

Note that we have as well 0.25=25100=25100=510=12.

Also, using decimal numbers instead of fractions gives, of course, the same result: 0.25=0.5, so 0.25=0.5=0.5.

Example 5: Taking the Square Root of a Mixed Number

Calculate ±361100.

Answer

To calculate this square root, let us first write 361100 as an improper fraction: 361100=3×100+61100=361100.

Now, we can apply the quotient rule: ±361100=±361100=±1910.

The answer is 1910,1910.

Let us summarize this explainer.

Key Points

  1. 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.
  2. 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.
  3. Product rule: 𝑎×𝑏=𝑎×𝑏.
  4. Quotient rule: 𝑎𝑏=𝑎𝑏.
  5. 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.
  6. To find the square root of a mixed number, write first the mixed number as an improper fraction and then apply the quotient rule.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.