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# 9.1: Evaluation of Square Roots

Difficulty Level: At Grade Created by: CK-12
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Have you ever loved a sport?

Miguel loves baseball. He is such a fan that he is volunteering all summer for the University. The University team, The “Wildcats” is an excellent team and Miguel is very excited to be helping out. He doesn’t even mind not being paid because he will get to see all of the games for free while he has the opportunity to learn more about baseball.

On the day of the first game, Miguel notices some big dark clouds as he rides his bike to the ball park. Sure enough as soon as the game is about to start, the rain begins. Like magic, a bunch of different people drag a huge tarp over the entire baseball infield. Miguel has never seen a tarp so big in his whole life.

He wonders how big the tarp actually is if it covers the entire infield. Miguel, being the fan that he is knows that the distance from one base to another, say \begin{align*}1^{st}\end{align*} to \begin{align*}2^{nd}\end{align*} is 90 feet. If the infield is in the shape of a square, then how many square feet is the infield? How can he be sure that his answer is correct?

Miguel begins to figure this out in his head.

Can you figure this out? Squaring numbers and finding their square roots is just one way to solve this problem. This Concept will teach you all about square roots and squaring. Pay close attention and at the end of the Concept you will be able to figure out the size of the tarp.

### Guidance

Think about a square for a minute. We can look at a square in a couple of different ways. First, we can look at just the outline of the square.

When you look at this square, you can see only the outside, but we all know that the side of a square can be measured and for a square to be a square it has to have four congruent sides.

Do you remember what congruent means?

It means exactly the same. So if a square has congruent sides, then they are the same length.

Now let’s say the side of a square is 3 units long. That means that each side of the square is 3 units long. Look at this picture of a square.

We call a number like this one a square number because it makes up a square. \begin{align*}3^2\end{align*} is represented in this square.

How many units make up the entire square?

If we count, we can see that this square is made up of 9 units. It is the same answer as \begin{align*}3^2\end{align*}, because \begin{align*}3^2\end{align*} is equal to 9.

Do you see a connection?

Think back to exponents, when we square a number, we multiply the number by itself. All squares have congruent side lengths, so the side length of a square multiplied by itself will tell you the number of units in the square.

We square the side length to find the number of units in the square.

This Concept is all about square roots. A square root is the number that we multiply by itself, or square, to get a certain result. In fact, if you square a number, when you take the square root of that number you will be back to the original number again.

Let’s think about the square that we just looked at. The dimensions of the square is \begin{align*}3 \times 3\end{align*}. We square the three to find the units in the square. The square root of the \begin{align*}3 \times 3\end{align*} square is 3. This is the value that we would multiply by itself.

We can find the square root of a number. How do we do this?

Finding the square root is the inverse operation of squaring a number. Inverse operations are simply the opposite of each other. Subtraction and addition are inverse operations, because one “undoes” the other. Similarly, squaring and finding the square root are inverse operations. When we find the square root, we look for the number that, times itself, will produce a given number.

We also use a symbol to show that we are looking for the square root of a number. Here is the symbol for square root.

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

If this were the problem, we would be looking for the square root of 9.

You could think of this visually as a square that has nine units in it. What would be the length of the side? It would be three.

You could also think of it using mental math to solve it. What number times itself is equal to nine. The answer is three.

When we find the square root of a number, we evaluate that square root.

\begin{align*}\sqrt{25}\end{align*}

This problem is asking us for the square root of 25. What number times itself is equal to 25? If you don’t know right away, you can think about this with smaller numbers.

\begin{align*}&3 \times 3 = 9\\ &4 \times 4 = 16\\ &5 \times 5 = 25\end{align*}

That’s it! The square root of 25 is 5.

We can also evaluate numbers where the square root is not a whole number.

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

To find the square root of seven, we can think about which two squares it is closest to.

\begin{align*}&2 \times 2 = 4\\ &3 \times 3 = 9\end{align*}

Seven is between four and nine, so we can say that the square root of seven is between 2 and 3.

Our answer would be that the \begin{align*}\sqrt{7}\end{align*} is between 2 and 3.

We can get a more exact number, but we aren’t going to worry about that for right now. Here is another one.

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

The square root of ten is between which two numbers?

\begin{align*}&3 \times 3 = 9\\ &4 \times 4 = 16\end{align*}

Our answer is that the \begin{align*}\sqrt{10}\end{align*} is between 3 and 4.

Now its time for you to try a few on your own. Evaluate each square root.

#### Example A

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

Solution:\begin{align*}6\end{align*}

#### Example B

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

Solution:\begin{align*}7\end{align*}

#### Example C

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

Solution: Between 3 and 4.

Here is the original problem once again.

Miguel loves baseball. He is such a fan that he is volunteering all summer for the University. The University team, The “Wildcats” is an excellent team and Miguel is very excited to be helping out. He doesn’t even mind not being paid because he will get to see all of the games for free while he has the opportunity to learn more about baseball.

On the day of the first game, Miguel notices some big dark clouds as he rides his bike to the ball park. Sure enough as soon as the game is about to start, the rain begins. Like magic, a bunch of different people drag a huge tarp over the entire baseball infield. Miguel has never seen a tarp so big in his whole life.

He wonders how big the tarp actually is if it covers the entire infield. Miguel, being the fan that he is knows that the distance from one base to another, say \begin{align*}1^{st}\end{align*} to \begin{align*}2^{nd}\end{align*} is 90 feet. If the infield is in the shape of a square, then how many square feet is the infield? How can he be sure that his answer is correct?

Miguel begins to figure this out in his head.

We can use what we know about squares to help us with this problem. We know that a square has four equal sides. This makes sense with baseball too. You want the distance from \begin{align*}1^{st}\end{align*} to \begin{align*}2^{nd}\end{align*} base to be the same as from \begin{align*}3^{rd}\end{align*} to Home. Therefore, if you know the distance from one base to another is 90 feet, then you know each distance from base to base.

However, Miguel wants to figure out the size of the tarp. He can do this by squaring the distance from \begin{align*}1^{st}\end{align*} to \begin{align*}2^{nd}\end{align*} base. This will give him the area of the square.

\begin{align*}90^2 = 90 \times 90 = 8100 \ square feet\end{align*}

This is the size of the tarp.

How can Miguel check the accuracy of his answer? He can do this by finding the square root of the area of the tarp. Remember that finding a square root is the inverse operation for squaring a number.

\begin{align*}\sqrt{8100}\end{align*}

To complete this, worry about the 81 and not the 8100. 81 is a perfect square. \begin{align*}9 \times 9 = 81\end{align*} so \begin{align*}90 \times 90 = 8100\end{align*}

\begin{align*}\sqrt{8100} = 90 \ ft\end{align*}

### 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.

### Guided Practice

Here is one for you to try on your own.

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

What is the square root of 64? What number times itself is 49? Let’s start where we left off with five.

\begin{align*}&6 \times 6 = 36\\ &7 \times 7 = 49\\ &8 \times 8 = 64\end{align*}

That’s it! The square root of 64 is 8.

### Video Review

Here is a video for review.

### Practice

Directions: Evaluate each square root.

1. \begin{align*}\sqrt{16}\end{align*}

2. \begin{align*}\sqrt{25}\end{align*}

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

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

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

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

7. \begin{align*}\sqrt{169}\end{align*}

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

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

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

Directions: Name the two values each square root is in between.

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

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

13. \begin{align*}\sqrt{30}\end{align*}

14. \begin{align*}\sqrt{40}\end{align*}

15. \begin{align*}\sqrt{50}\end{align*}

16. \begin{align*}\sqrt{62}\end{align*}

17. \begin{align*}\sqrt{70}\end{align*}

18. \begin{align*}\sqrt{101}\end{align*}

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

20. \begin{align*}\sqrt{15}\end{align*}

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