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Evaluation of Perfect Square Roots

Understand that perfect square roots are whole numbers squared.

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Evaluation of Perfect Square Roots
License: CC BY-NC 3.0

Miguel was helping his little brother, Luis, cut several sheets of paper into 4 squares each. The sheets measure 9 inches x 9 inches. Luis insisted that all the squares be "perfect." This reminded Miguel of a math lesson on the difference between "perfectly square," and "perfect squares." Luis wanted cut outs that were "perfectly square." Miguel's homework involved "perfect squares." Do you know the difference?

In this concept, you will learn about perfect square roots.

Evaluating Perfect Square Roots

square has four equal sides, meaning it is "perfectly square." This means it is not a rectangle or a parallelogram or any other 4-sided figure. 

If you know the side lengths of a square and want to find the area, you simply "square" the sides.

The square root is the opposite calculation. If you know the area of the square and want to find the sides, you determine the root by using your knowledge of the multiplication tables.

A square root is the number that, times itself, produces a given number. Perfect squares are numbers whose square roots are whole numbers.

These numbers are perfect squares: 4, 9, 16, 25, 36

When you look at their images, you can see why.

License: CC BY-NC 3.0

These numbers are perfect squares because their square roots are whole numbers.

Let’s examine one.


First, you know that 4 x 4 equals 16.

The answer is 4.

4 is the square root of 16, and because 4 is a whole number, we say that 16 is a perfect square.

Perfect squares have the easiest square roots to find because they are whole numbers. Most numbers are not perfect squares.

License: CC BY-NC 3.0

What is the square root of 5?

First, the diagram shows that 4, 5, and 9 are all "perfectly square," meaning that in each square, the sides are equal.

Next, the square root of 5 cannot be easily determined because 5 does not appear in the multiplication tables. You cannot easily identify a number to multiply times itself to get 5 as the answer. 

Then, you know that 5 is not a perfect square. Its square root is not a whole number but is somewhere between 2 and 3. It falls between the perfect squares of 4 and 9. 

Next, using a calculator, you can solve the equation.

The answer is is roughly 2.236067978

That is definitely not a perfect square! But your estimate of between 2 and 3 was correct.


Example 1

Earlier, you were given a problem about Miguel and Luis, who are cutting squares.

If the boys are using paper that measures 9 inches by 9 inches and are cutting it into four pieces, are they cutting perfect squares or just pieces that are perfectly square?

License: CC BY-NC 3.0

First, since the sheets of paper measure 9 inches x 9 inches, and 9 is a whole number, you know that  and 81 is a perfect square. The uncut sheets are perfect squares.

Next, when the paper is cut into four squares, each smaller square has sides that measure 4.5 inches x 4.5 inches.

Then, since 4.5 is not a whole number, Luis's cut outs are not perfect squares.

The answer is that Luis's squares are perfectly square, but they are not perfect squares. 

Example 2

Is 169 a perfect square? 


First, think about what number times itself could be equal to 169.


Next, let's try 13.


The answer is 13. The square root of 169 is 13. Since 13 is a whole number, 169 is a perfect square.

Find the following square roots and identify the perfect squares.

Example 3


First, think of what number times itself might equal 121.


Next, try 11.


The answer is 11. The square root of 121 equals 11. Since 11 is a whole number, 121 is a perfect square.

Example 4


First, think of the multiplication tables.



Next, 74 lies somewhere between 64 and 81.

Then, lies somewhere between 8 and 9.

Use a calculator and round your answer to one decimal.

The answer is 8.6 and this root is not a whole number. 74 is not a perfect square.

Example 5


First, think of the perfect squares you know.

6 x 6 = 36

7 x 7 = 49

Then, 40 is not a perfect square.

The answer is is somewhere between 6 and 7. You can use a calculator to find the exact root.


Evaluate each square root.

  1. What do all of these square roots have in common?

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.2.


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Perfect Square

A perfect square is a number whose square root is an integer.


A square is a polygon with four congruent sides and four right angles.

Square Number

A square number or perfect square is a whole number that is the square of another integer. For example, 36 is a square number because 6 \cdot 6 = 36.

Square Root

The square root of a term is a value that must be multiplied by itself to equal the specified term. The square root of 9 is 3, since 3 * 3 = 9.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0

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