9.1: Square Roots
Introduction
Delay of Game
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 does the infield cover? 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 lesson will teach you all about square roots and squaring. Pay close attention and at the end of the lesson you will be able to figure out the size of the tarp.
What You Will Learn
In this lesson you will learn the following skills:
- Evaluate square roots.
- Recognize perfect squares.
- Evaluate radical expressions.
- Solve equations using square roots.
Teaching Time
I. Evaluate Square Roots
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. These integers, which are equal to a different integer squared, are called "perfect squares".
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 lesson 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, and then take the square root of the result, 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 are \begin{align*}3 \times 3\end{align*}. We square the three to find the units in the square shape. The square root of the \begin{align*}3 \times 3\end{align*} square is 3.
We can practice finding 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. To 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. Let’s look at an example.
Example
\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.
Example
\begin{align*}\sqrt{49}\end{align*}
What is the square root of 49? What number times itself is 49? Let’s start where we left off with five.
\begin{align*}&5 \times 5 = 25\\ &6 \times 6 = 36\\ &7 \times 7 = 49\end{align*}
That’s it! The square root of 49 is 7.
We can also evaluate numbers where the square root is not a whole number.
Let’s look at an example
Example
\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.
Example
\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.
Working with square roots that are whole numbers is a lot easier! Let’s look at what we call a perfect square.
II. Recognize Perfect Squares
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. For example, the square root of 5 is roughly 2.236067978! But let’s not worry about that right now.
9A. Lesson Exercises
Evaluate each perfect square by finding its square root.
- 64
- 1
- 100
Go over your answers with a friend and then continue with the next section.
III. Evaluate Radical Expressions
Sometimes, we can have an expression with a radical in it.
What is a radical?
A radical is the name of the sign that tells us that we are looking for a square root. We can call this “a radical.” Here is a radical symbol.
\begin{align*}\sqrt{y}\end{align*}
Here we would be looking for the square root of \begin{align*}y\end{align*}.
What is a radical expression?
Remember that an expression is a number sentence that contains numbers, operations and now radicals. Just as we can have expressions without radicals, we can have expressions with them too.
Here is an example of a radical expression.
Example
\begin{align*}2 \cdot \sqrt{4} + 7\end{align*}
Here we have two times the square root of four plus seven.
That’s a great question! We can evaluate this expression by using the order of operations.
P parentheses
E exponents (square roots too)
MD multiplication/division in order from left to right
AS addition/subtraction in order from left to right
According to the order of operations, we evaluate the square root of 4 first.
\begin{align*}\sqrt{4} = 2\end{align*}
Next, we substitute that value into the expression.
\begin{align*}2 \cdot 2 + 7\end{align*}
Next, we complete multiplication/division in order from left to right.
\begin{align*}2 \times 2 = 4\end{align*}
Substitute that given value.
\begin{align*}4 + 7 = 11\end{align*}
The answer is 11.
Let’s look at another one.
Example
\begin{align*}\sqrt{4} \cdot \sqrt{16} - 3\end{align*}
Here we have two radicals in the expression. We can work the same way, by using the order of operations.
Evaluate the radicals first.
\begin{align*}\sqrt{4} = 2\!\\ \sqrt{16} = 4\end{align*}
Substitute these values into the expression.
\begin{align*}2 \cdot 4 - 3\end{align*}
Next, we complete multiplication/division in order from left to right.
\begin{align*}2 \times 4 = 8\end{align*}
Finally, we complete the addition/subtraction in order from left to right.
\begin{align*}8 - 3 = 5\end{align*}
The answer is 5.
9B. Lesson Exercises
Evaluate each radical expression.
- \begin{align*}6 + \sqrt{9} - \sqrt{49} + 5\end{align*}
- \begin{align*}\sqrt{64} \div \sqrt{4} + 13\end{align*}
- \begin{align*}6(7) + \sqrt{121} - 3\end{align*}
Check your answers carefully with a friend. Did you follow the order of operations?
IV. Solve Equations Using Square Roots
If we know that the square of a number is equal to a product, then we can write this equation.
\begin{align*}y = x^2\end{align*}
This equation expresses that if we multiply the value of \begin{align*}x\end{align*} by itself, we will end up with the value of \begin{align*}y\end{align*}.
We can say that \begin{align*}x\end{align*} is the square root of \begin{align*}y\end{align*}.
We can use this information to solve equations involving square roots.
Example
\begin{align*}x^2=81\end{align*}
To solve this equation, we want to figure out the value of \begin{align*}x\end{align*}. To do this, we can take the square root of both sides of the equation.
Why would we do this?
Think about it. The \begin{align*}x\end{align*} is being squared. We want to get the variable alone. To do this, we need to perform the inverse operation. The opposite of squaring a number is finding the square root of the number. Therefore, if we take the square root of both sides of the equation, then we will get \begin{align*}x\end{align*} alone.
\begin{align*}\sqrt{x^2} = \sqrt{81}\end{align*}
Next, we can cancel the square and the square root. They are inverses of each other and they cancel each other out.
\begin{align*}\bcancel{\sqrt{x^2}} = x\end{align*}
Now we find the square root of 81. 81 is a perfect square, and the square root of 81 is 9.
\begin{align*}x=9\end{align*}
This is our answer.
Sometimes, you will have a problem that is a little more complicated. Take a look.
Example
\begin{align*}x^2+3=12\end{align*}
We want to find the value of \begin{align*}x\end{align*}. First, notice that we have a two step equation. One of the operations involves an exponent and the other is addition.
When solving an equation by cancelling, you are "undoing" all of the present operations. Along with "undoing" all of the operations, you also "undo" the ORDER of operations. Because of this, let’s start by subtracting three from both sides.
\begin{align*}x^2+3-3&=12-3\\ x^2&=9\end{align*}
Now we want to get \begin{align*}x\end{align*} alone. To do this, we take the square root of both sides of the equation.
\begin{align*}\bcancel{\sqrt{x^2}} &= \sqrt{9}\\ x&=3\end{align*}
This is our answer.
Sometimes, the equation with have a square root in it and we have to work with that.
Example
\begin{align*}\sqrt{x-1} = 8\end{align*}
Wow! Here we have a variable and a number in a radical. Let’s get rid of the radical first.
To get cancel out the square root of a number, we use the inverse operation. We square both sides of the equation.
\begin{align*}\left ( \sqrt{x-1} \right )^2 = 8^2\end{align*}
The square and the square root cancel each other out.
\begin{align*}(\bcancel{\sqrt{x-1})^2} &= 8^2\\ x-1 &= 64\end{align*}
Now we can solve for \begin{align*}x\end{align*} quite easily. Begin by adding 1 to both sides of the equation.
\begin{align*}x-1+1&=64+1\\ x&=65\end{align*}
This is our answer.
Now let’s return to the introductory problem and use what we have learned to help us to solve this problem.
Real–Life Example Completed
Delay of Game
Here is the original problem once again. Reread the problem and underline any of the important information.
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 does the infield cover? 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*}
Miguel’s answer checks out.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- 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 a given starting number.
- Perfect Square
- a number with square roots that are whole numbers.
- Radical
- the symbol that lets us know that we are looking for a square root.
- Radical Expression
- an expression with numbers, operations and radicals in it.
Technology Integration
James Sousa, Introduction to Square Roots
Time to 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*}
Directions: Evaluate each radical expression.
21. \begin{align*}2 + \sqrt{9} + 15-2\end{align*}
22. \begin{align*}3 \cdot 4 + \sqrt{169}\end{align*}
23. \begin{align*}\sqrt{16} \cdot \sqrt{25} + \sqrt{36}\end{align*}
24. \begin{align*}\sqrt{81} \cdot 12 + 12\end{align*}
25. \begin{align*}\sqrt{36} + \sqrt{47} - \sqrt{16}\end{align*}
Directions: Solve each equation.
26. \begin{align*}x^2=9\end{align*}
27. \begin{align*}x^2=49\end{align*}
28. \begin{align*}x^2=256\end{align*}
29. \begin{align*}\sqrt{x+1} = 10\end{align*}
30. \begin{align*}x^2+5=41\end{align*}