3.5 Square Roots

Narrated lesson: Square Roots

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Simplify Expressions with Square Roots

To start this section, we need to review some important vocabulary and notation.

Remember that when a number $n$ is multiplied by itself, we can write this as $n^{2},$ which we read aloud as $“ n squared.”$ For example, $8 squared$ is read as $“8 squared.”$

We call $64$ the square of $8$ because $8 squared equals 64.$ Similarly, $121$ is the square of $11,$ because $11 squared equals 121.$

Modeling Squares

Do you know why we use the word square? If we construct a square with three tiles on each side, the total number of tiles would be nine.

A square is shown with 3 tiles on each side. There are a total of 9 tiles in the square.

This is why we say that the square of three is nine.

$3 squared equals 9$

The number $9$ is called a perfect square because it is the square of a whole number.

The chart shows the squares of the counting numbers $1$ through $15 .$ You can refer to it to help you identify the perfect squares.

A table with two columns is shown. The first column is labeled “Number” and has the values: n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second column is labeled “Square” and has the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.

What happens when you square a negative number?

$\begin{matrix} {(−8)}^{2} & = (−8) (−8) \\ = 64 \end{matrix}$

When we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive.

The chart shows the squares of the negative integers from $−1$ to $−15.$

A table is shown with 2 columns. The first column is labeled “Number” and contains the values: n, negative 1, negative 2, negative 3, negative 4, negative 5, negative 6, negative 7, negative 8, negative 9, negative 10, negative 11, negative 12, negative 13, negative 14, and negative 15. The next column is labeled “Square” and contains the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.

Did you notice that these squares are the same as the squares of the positive numbers?

Square Roots

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because $10 squared equals 100,$ we say $100$ is the square of $10 .$ We can also say that $10$ is a square root of $100 .$

Notice $open parenthesis −10 close parenthesis squared equals 100$ also, so $−10$ is also a square root of $100 .$ Therefore, both $10$ and $−10$ are square roots of $100 .$

So, every positive number has two square roots: one positive and one negative.

What if we only want the positive square root of a positive number? The radical sign, $\sqrt{},$ stands for the positive square root. The positive square root is also called the principal square root.

We can also use the radical sign for the square root of zero. Because $0^{2} = 0, \sqrt{0} = 0 .$ Notice that zero has only one square root.

The chart shows the square roots of the first $15$ positive perfect square numbers.

A table is shown with 2 columns. The first column contains the values: square root of 1, square root of 4, square root of 9, square root of 16, square root of 25, square root of 36, square root of 49, square root of 64, square root of 81, square root of 100, square root of 121, square root of 144, square root of 169, square root of 196, and square root of 225. The second column contains the values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.

Every positive number has two square roots and the radical sign indicates the positive one. We write $\sqrt{100} = 10 .$ If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, $- \sqrt{100} = −10 .$

Square Root of a Negative Number

Can we simplify $the square root of −25 ?$ Is there a number whose square is $−25 ?$

$open parenthesis close parenthesis squared equals −25 ?$

None of the numbers that we have dealt with so far have a square that is $−25.$ Why? Any positive number squared is positive, and any negative number squared is also positive. All the numbers we work with are called the real numbers. So we say there is no real number equal to $the square root of −25.$ If we are asked to find the square root of any negative number, we say that the solution is not a real number.

Example 3

Simplify: ⓐ $the square root of −169$ ⓑ $- \sqrt{121} .$

Show solution

ⓐ There is no real number whose square is $−169.$ Therefore, $the square root of −169$ is not a real number.

ⓑ The negative is in front of the radical sign, so we find the opposite of the square root of $121 .$

	$- \sqrt{121}$
The negative is in front of the radical.	$minus 11$

Square Roots and the Order of Operations

When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.

Example 4

Simplify: ⓐ $\sqrt{25} + \sqrt{144}$ ⓑ $\sqrt{25 + 144} .$

Show solution

ⓐ Use the order of operations.
	$\sqrt{25} + \sqrt{144}$
Simplify each radical.	$5 plus 12$
Add.	$17$

ⓑ Use the order of operations.
	$\sqrt{25 + 144}$
Add under the radical sign.	$the square root of 169$
Simplify.	$13$

Notice the different answers in parts ⓐ and ⓑ of Example 4. It is important to follow the order of operations correctly. In ⓐ, we took each square root first and then added them. In ⓑ, we added under the radical sign first and then found the square root.

Estimate Square Roots

So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.

A table is shown with 2 columns. The first column is labeled “Number” and contains the values: 4, 5, 6, 7, 8, 9. The second column is labeled “Square root” and contains the values: square root of 4 equals 2, square root of 5, square root of 6, square root of 7, square root of 8, square root of 9 equals 3.

We might conclude that the square roots of numbers between $4$ and $9$ will be between $2$ and $3,$ and they will not be whole numbers. Based on the pattern in the table above, we could say that $the square root of 5$ is between $2$ and $3 .$ Using inequality symbols, we write

$2 less than the square root of 5 less than 3$

Example 5

Estimate $the square root of 60$ between two consecutive whole numbers.

Show solution

Think of the perfect squares closest to $60 .$ Make a small table of these perfect squares and their square roots.

$Locate 60 between two consecutive perfect squares.$	$49 less than 60 less than 64$
$the square root of 60 is between their square roots.$	$7 less than the square root of 60 less than 8$

Approximate Square Roots with a Calculator

There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the or $\sqrt{x}$ key on your calculator. You will need to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator’s display. The symbol for an approximation is $\approx$ and it is read approximately.

Suppose your calculator has a $10-digit$ display. Using it to find the square root of $5$ will give $2.236067977.$ This is the approximate square root of $5 .$ When we report the answer, we should use the “approximately equal to” sign instead of an equal sign.

$\sqrt{5} \approx 2.236067977$

You will seldom use this many digits for applications in algebra. So, if you wanted to round $the square root of 5$ to two decimal places, you would write

$\sqrt{5} \approx 2.24$

How do we know these values are approximations and not the exact values? Look at what happens when we square them.

$\begin{matrix} {2.236067977}^{2} & = & 4.999999997764872529 \\ {2.24}^{2} & = & 5.0176 \end{matrix}$

The squares are close, but not exactly equal, to $5 .$

Section Exercises

Practice Makes Perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

$the square root of 36$

Show solution

$the square root of 4$

$the square root of 64$

Show solution

$the square root of 144$

$- \sqrt{4}$

Show solution

−2

$- \sqrt{100}$

$- \sqrt{1}$

Show solution

−1

$- \sqrt{121}$

$the square root of −121$

Show solution

not a real number

$the square root of −36$

$the square root of −9$

Show solution

not a real number

$the square root of −49$

$\sqrt{9 + 16}$

Show solution

$\sqrt{25 + 144}$

$\sqrt{9} + \sqrt{16}$

Show solution

$\sqrt{25} + \sqrt{144}$

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

$the square root of 70$

Show solution

$8 less than the square root of 70 less than 9$

$the square root of 55$

$the square root of 200$

Show solution

$14 less than the square root of 200 less than 15$

$the square root of 172$

Approximate Square Roots with a Calculator

In the following exercises, use a calculator to approximate each square root and round to two decimal places.

$the square root of 19$

Show solution

4.36

$the square root of 21$

$the square root of 53$

Show solution

7.28

$the square root of 47$

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ⓑ
	$the square root of 121$
Since $11 squared equals 121$	$11$

ⓐ
	$- \sqrt{9}$
The negative is in front of the radical sign.	$minus 3$

ⓑ
	$- \sqrt{144}$
The negative is in front of the radical sign.	$minus 12$

	$the square root of 17$
Use the calculator square root key.	$4.123105626$
Round to two decimal places.	$4.12$
	$\sqrt{17} \approx 4.12$