# 9.2: Evaluation of Perfect Square Roots

**At Grade**Created by: CK-12

Have you ever examined a baseball diamond carefully? Look at this diagram.

This diagram shows that the distance from one base to another is 90 feet. Each base has this same measurement. The fact that the distance is equal from one base to another helps to define it as a square. When two sides of a square are equal, they have an area that forms a special kind of square root.

Take a look.

\begin{align*}A = s^2\end{align*}

\begin{align*}A = 90^2 = 8100 \ sq. ft.\end{align*}

Here is a square with a side length of 90 feet. When we multiply this measurement by itself, we have an area of 8100 square feet.

Hmmm. Can you figure out the square root of that? This is an example of a perfect square root. Do you know what that means?

**Use this Concept to learn about perfect square roots. Then we will return to this problem at the end of the Concept.**

### Guidance

We have seen that a square root is the number that, times itself, produces a given number. *Perfect squares***are numbers whose square roots are whole numbers.** The numbers below are perfect squares. Take a good look at them. Can you guess why?

4, 9, 16, 25, 36

Look at them as images.

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

Let’s examine each one. What is the square root of 4? In other words, what number, when multiplied by itself, equals 4? \begin{align*}2 \times 2\end{align*} equals 4, so 2 is the square root of 4. Because 2 is a whole number, we say that 4 is a perfect square.

What is the square root of 9? We have already finished that one, it is 3.

What is the square root of 16? \begin{align*}4 \times 4 = 16\end{align*}.

What is the square root of 25? \begin{align*}5 \times 5 = 25\end{align*}.

What is the square root of 36? \begin{align*}6 \times 6 = 36\end{align*}.

**When we look at the square root of each of these perfect squares, we end up with a whole number. Perfect squares are the easiest square roots to find because they are whole numbers.**

**Most numbers are not perfect squares.**

The square root of 5 is roughly 2.236067978! But let’s not worry about that right now.

Evaluate each perfect square by finding its square root.

#### Example A

\begin{align*}64\end{align*}

**Solution: \begin{align*}8\end{align*}**

#### Example B

\begin{align*}16\end{align*}

**Solution: \begin{align*}4\end{align*}**

#### Example C

\begin{align*}100\end{align*}

**Solution: \begin{align*}10\end{align*}**

Here is the original problem once again.

This diagram shows that the distance from one base to another is 90 feet. Each base has this same measurement. The fact that the distance is equal from one base to another helps to define it as a square. When two sides of a square are equal, they have an area that forms a special kind of square root.

Take a look.

\begin{align*}A = s^2\end{align*}

\begin{align*}A = 90^2 = 8100 \ sq. ft.\end{align*}

Here is a square with a side length of 90 feet. When we multiply this measurement by itself, we have an area of 8100 square feet.

Hmmm. Can you figure out the square root of that? This is an example of a perfect square root. Do you know what that means?

81 is a perfect square root because 9 x 9 = 81.

We can think about 8100 in the same way.

90 x 90 = 8100

**A baseball diamond is an example of a perfect square.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Square
- a four sided figure with congruent sides.

- Congruent
- exactly the same

- Square Number
- a number of units which makes a perfect square.

- Square root
- a number that when multiplied by itself equals the square of the number.

- Perfect Square
- square roots that are whole numbers.

### Guided Practice

Here is one for you to try on your own.

Evaluate square root of 169.

**Answer**

To figure this out, we have to think of what number times itself could be equal to 169?

We know that 12 x 12 = 144.

Let's try 13.

13 x 13 = 169.

**Our answer is 13. It is a perfect square root.**

### Video Review

Here is a video for review.

- This is a Khan Academy video on understanding square roots.

### Practice

Directions: Evaluate each square root.

1. \begin{align*}\sqrt{36}\end{align*}

2. \begin{align*}\sqrt{49}\end{align*}

3. \begin{align*}\sqrt{64}\end{align*}

4. \begin{align*}\sqrt{81}\end{align*}

5. \begin{align*}\sqrt{100}\end{align*}

6. \begin{align*}\sqrt{121}\end{align*}

7. \begin{align*}\sqrt{144}\end{align*}

8. \begin{align*}\sqrt{196}\end{align*}

9. \begin{align*}\sqrt{400}\end{align*}

10. \begin{align*}\sqrt{900}\end{align*}

11. \begin{align*}\sqrt{1000}\end{align*}

12. \begin{align*}\sqrt{225}\end{align*}

13. \begin{align*}\sqrt{256}\end{align*}

14. \begin{align*}\sqrt{324}\end{align*}

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

### Image Attributions

Here you'll learn to recognize perfect square roots.