Square Root: Perfect Squares and Estimating

A square root of a number is a value that you multiply by itself to get that number. The square root of 25 is 5, because $5 \times 5 = 25$ . We write this as $\sqrt{25} = 5$ . You can find exact square roots of perfect squares and estimate all the others.

Before this lesson: You should know what squaring means from Exponents. It also helps to know how to multiply integers and work with fractions.

Extra practice: The Pre-Algebra Unit 4 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

What Is a Square Root?

Squaring a number means multiplying it by itself: $5^2 = 5 \times 5 = 25$ . A square root goes the other way. It asks, “What number times itself gives 25?”

Think of a square made of small tiles. If the square has 25 tiles, each side is 5 tiles long. The area is 25, and the square root of the area is the side length.

A square made of 5 rows of 5 small squares, for an area of 25 square units. Each side is 5 units long, so the square root of 25 is 5. — The square root of an area is the side length of the square: √25 = 5.

Both 5 and −5 square to 25, since $(-5) \times (-5) = 25$ . But the symbol √ always means the positive square root. This is called the principal square root.

The Square Root Rule and Perfect Squares

\sqrt{a} = b \quad \text{means} \quad b \times b = a \text{ and } b \text{ is not negative}

The number under the √ symbol is called the radicand. The √ symbol itself is called the radical sign.

A perfect square is a number made by squaring a whole number. Perfect squares have whole number square roots. Learn these by heart, because you will use them all the time.

Square	Square root	Square	Square root
$1^2 = 1$	$\sqrt{1} = 1$	$7^2 = 49$	$\sqrt{49} = 7$
$2^2 = 4$	$\sqrt{4} = 2$	$8^2 = 64$	$\sqrt{64} = 8$
$3^2 = 9$	$\sqrt{9} = 3$	$9^2 = 81$	$\sqrt{81} = 9$
$4^2 = 16$	$\sqrt{16} = 4$	$10^2 = 100$	$\sqrt{100} = 10$
$5^2 = 25$	$\sqrt{25} = 5$	$11^2 = 121$	$\sqrt{121} = 11$
$6^2 = 36$	$\sqrt{36} = 6$	$12^2 = 144$	$\sqrt{144} = 12$

A negative number has no square root in the numbers you know. No number times itself gives a negative answer. So $\sqrt{-9}$ is not −3, because $(-3) \times (-3) = 9$ .

Square Roots of Fractions and Decimals

For a fraction, take the square root of the top and the bottom. This works when both are perfect squares.

\sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}

For a decimal, look for a decimal that times itself gives the number. For example, $\sqrt{0.36} = 0.6$ because $0.6 \times 0.6 = 0.36$ . The answer has half as many decimal places as the radicand.

How to Estimate a Square Root

Most numbers are not perfect squares. Their square roots are not whole numbers. You can still estimate them in three steps.

Find the perfect square just below the number and the one just above it.
Take their square roots. The answer is between these two whole numbers.
Check which perfect square is closer. The answer is closer to that square’s root.

For $\sqrt{50}$ , the nearby perfect squares are 49 and 64. So $\sqrt{50}$ is between 7 and 8. Since 50 is very close to 49, the answer is just a bit more than 7.

Number line from 6 to 9. The perfect squares 36, 49, 64 and 81 are marked above 6, 7, 8 and 9. A point for the square root of 50 sits just to the right of 7, because 50 is just a little more than 49. — 50 is between the perfect squares 49 and 64, and much closer to 49, so √50 is a little more than 7.

A calculator gives $\sqrt{50} \approx 7.07$ . The symbol ≈ means “about equal to.”

Worked Examples

Example 1: Square root of a perfect square

Find $\sqrt{144}$ .

Step 1. Ask what number times itself gives 144.

12 \times 12 = 144

Step 2. Try 12.

Answer: $\sqrt{144} = 12$

Example 2: Square root of a fraction

Find $\sqrt{\frac{16}{25}}$ .

\sqrt{16} = 4

Step 1. Take the square root of the top.

\sqrt{25} = 5

Step 2. Take the square root of the bottom.

Step 3. Check: $\frac{4}{5} \times \frac{4}{5} = \frac{16}{25}$ .

Answer: $\sqrt{\frac{16}{25}} = \frac{4}{5}$

Example 3: Square root of a decimal

Find $\sqrt{0.09}$ .

\sqrt{9} = 3

Step 1. Ignore the decimal for a moment. The square root of 9 is 3, so the answer uses the digit 3.

Step 2. The radicand has 2 decimal places, so the answer has 1. Try 0.3.

0.3 \times 0.3 = 0.09

Step 3. Check: multiply 0.3 by itself.

Answer: $\sqrt{0.09} = 0.3$

Example 4: Estimate a square root

Estimate $\sqrt{30}$ to one decimal place.

Step 1. The perfect squares around 30 are 25 and 36.

Step 2. So $\sqrt{30}$ is between $\sqrt{25} = 5$ and $\sqrt{36} = 6$ .

Step 3. 30 is 5 away from 25 and 6 away from 36. That is close to the middle, so try 5.5.

Step 4. Check: $5.5 \times 5.5 = 30.25$ . That is very close to 30.

Answer: $\sqrt{30} \approx 5.5$ . A calculator gives 5.48.

Example 5: Word problem with a square garden

A square garden has an area of 64 square feet. How long is each side? How much fence goes around it?

Step 1. The area of a square is side × side. So the side is $\sqrt{64}$ .

8 \times 8 = 64

Step 2. Find the number that times itself gives 64.

\sqrt{64} = 8

Step 3. So the square root of 64 is 8. Each side is 8 feet.

4 \times 8 = 32

Step 4. A square has 4 equal sides. Multiply to find the fence length in feet.

Answer: Each side is 8 feet, and the fence is 32 feet long.

Common Mistakes with Square Roots

Mistake	Example of the error	Fix
Dividing by 2	$\sqrt{16} = 8$	Ask what times itself gives 16. $4 \times 4 = 16$ , so $\sqrt{16} = 4$ .
Giving a negative answer for √	$\sqrt{25} = -5$	The √ symbol means the positive root. $\sqrt{25} = 5$ .
Wrong decimal places	$\sqrt{0.4} = 0.2$	Check: $0.2 \times 0.2 = 0.04$ , not 0.4. So $\sqrt{0.04} = 0.2$ , and $\sqrt{0.4} \approx 0.63$ .
Root of a negative number	$\sqrt{-9} = -3$	$(-3) \times (-3) = 9$ , not −9. No number you know squares to −9.
Splitting a sum	$\sqrt{9 + 16} = 3 + 4 = 7$	Add first: $\sqrt{9 + 16} = \sqrt{25} = 5$ .

Square Roots and Squaring

Squaring and taking a square root undo each other. If you square a number and then take the square root, you get back to where you started.

\sqrt{7^2} = \sqrt{49} = 7 \qquad \left(\sqrt{49}\right)^2 = 7^2 = 49

You will use this idea to solve equations like $x^2 = 49$ in Algebra 1. The next lesson covers cube roots, which undo cubing.

Practice Problems

Find $\sqrt{36}$ .
Find $\sqrt{121}$ .
Find $\sqrt{400}$ .
Find $\sqrt{\frac{9}{49}}$ .
Find $\sqrt{0.64}$ .
Between which two whole numbers is $\sqrt{70}$ ? Which one is it closer to?
Find $\sqrt{36 + 64}$ .
A square tile has an area of 2.25 square feet. How long is each side?
Is 50 a perfect square? Explain.

Show answers

6 \times 6 = 36

Find the number that times itself gives 36.

\sqrt{36} = 6

Write the square root.

11 \times 11 = 121

Find the number that times itself gives 121.

\sqrt{121} = 11

Write the square root.

20 \times 20 = 400

Find the number that times itself gives 400.

\sqrt{400} = 20

Write the square root.

\frac{\sqrt{9}}{\sqrt{49}} = \frac{3}{7}

Take the square root of the top and the bottom.

0.8 \times 0.8 = 0.64

Find the number that times itself gives 0.64.

\sqrt{0.64} = 0.8

Write the square root.

70 is between 64 and 81, so $\sqrt{70}$ is between 8 and 9. It is closer to 8, because 70 is 6 away from 64 and 11 away from 81. (It is about 8.4.)

\sqrt{36 + 64} = \sqrt{100}

Add first.

\sqrt{100} = 10

Take the square root.

The answer is not 6 + 8.

1.5 \times 1.5 = 2.25

Find the number that times itself gives 2.25.

\sqrt{2.25} = 1.5

Write the square root.

Answer: Each side is 1.5 feet.

No.

7^2 = 49

Square 7.

8^2 = 64

Square 8.

50 is between them. No whole number squares to 50.

Want more practice? The Pre-Algebra Unit 4 Practice Pack has 35 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Square Root FAQ

What is the square root of 2?

The square root of 2 is about 1.414. Its decimal never ends and never repeats, so we often leave it as $\sqrt{2}$ .

Can a square root be negative?

Both 5 and −5 square to 25. The √ symbol means only the positive one, so $\sqrt{25} = 5$ . To show the negative root, write $-\sqrt{25} = -5$ .

What is the square root of 0?

The square root of 0 is 0, because $0 \times 0 = 0$ .

Is a square root always smaller than the number?

No. For numbers bigger than 1, the square root is smaller. But $\sqrt{0.25} = 0.5$ , which is bigger than 0.25. And $\sqrt{1} = 1$ .

Summary

A square root of a number times itself gives the number: $\sqrt{25} = 5$ because $5 \times 5 = 25$ .
The √ symbol means the positive (principal) square root.
Perfect squares like 1, 4, 9, 16 and 25 have whole number square roots.
For a fraction, take the root of the top and the bottom. For a decimal, the answer has half as many decimal places.
To estimate, find the perfect squares on each side of the number.
Negative numbers have no square root in the numbers you know yet.

Previous lesson: Exponents

Next lesson: Cube Roots

Back to the Pre-Algebra Index