Square Roots Examples: 41 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Square Roots.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The non-negative number $b$ such that $b^2 = a$ , written $\sqrt{a} = b$ — the inverse of squaring.

$\sqrt{25}$ asks: what number times itself equals 25? Answer: 5.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A square root asks for the side length of a square with a given area.

Common stuck point: The procedure for square roots is the easy part; the trap is dividing by 2 instead of finding a self-product. Asking "What number multiplied by itself gives the radicand?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: What number multiplied by itself gives the radicand?

Worked Examples

Example 1

easy

Find

\sqrt{144}

Answer

12

First step

Recall that a square root asks for the positive number whose square equals the original number.

Full solution

2
Ask: what number multiplied by itself gives 144?
3
Test: $12 \times 12 = 144$ . So $\sqrt{144} = 12$ .

The square root of a number

n

is the value that, when multiplied by itself, produces

n

. Memorizing perfect squares (1, 4, 9, 16, 25, ..., 144) makes these computations fast.

Example 2

medium

A square has an area of

196

cm². What is the side length of the square?

Example 3

medium

A right triangle has legs

6

and

8

. Find the hypotenuse.

Example 4

hard

Rationalize

\dfrac{1}{\sqrt{5}+1}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Simplify

\sqrt{200}

Example 2

easy

A square garden has an area of 144 square feet. What is the side length?

Example 3

easy

Find

\sqrt{49}

Example 4

easy

Find

\sqrt{81}

Example 5

easy

Find

\sqrt{100}

Example 6

easy

Find

\sqrt{1}

Example 7

easy

Find

\sqrt{0}

Example 8

easy

Find

\sqrt{144}

Example 9

easy

Find

\sqrt{\frac{1}{4}}

Example 10

easy

Find

\sqrt{25} + \sqrt{16}

Example 11

medium

Estimate

\sqrt{50}

to the nearest whole number.

Example 12

medium

Simplify

\sqrt{72}

to the form

a\sqrt{b}

Example 13

medium

A square garden has area

169

square feet. What is its side length?

Example 14

medium

Evaluate

\sqrt{36} \times \sqrt{4}

Example 15

medium

Solve

x^2 = 81

for all real

x

Example 16

medium

Estimate

\sqrt{30}

to one decimal place.

Example 17

medium

Simplify

\sqrt{\frac{49}{64}}

Example 18

medium

The diagonal of a square satisfies

d^2 = 2s^2

. If

s = 5

, find

d

in simplest radical form.

Example 19

medium

Evaluate

\sqrt{64} - \sqrt{9}

Example 20

challenge

For how many integers

n

with

1 \le n \le 100

\sqrt{n}

an integer?

Example 21

challenge

Simplify

\sqrt{12} + \sqrt{27}

to the form

a\sqrt{b}

Example 22

challenge

Between which two consecutive integers does

\sqrt{200}

lie, and which is it closer to?

Example 23

easy

Find

\sqrt{225}

Example 24

easy

Find

\sqrt{121}

Example 25

easy

True or false:

\sqrt{36}=18

Example 26

easy

Find

\sqrt{0.25}

Example 27

medium

Simplify

\sqrt{98}

Example 28

medium

Simplify

\sqrt{50}+\sqrt{8}

Example 29

medium

Solve

x^2=144

for all real

x

Example 30

medium

Simplify

\sqrt{45}-\sqrt{20}

Example 31

medium

Rationalize the denominator:

\dfrac{6}{\sqrt{3}}

Example 32

medium

Between which two consecutive integers does

\sqrt{60}

lie?

Example 33

hard

Simplify

\sqrt{12}\cdot\sqrt{75}

Example 34

hard

Solve

\sqrt{x+5}=4

for

x

Example 35

hard

Compute

(\sqrt{7}+\sqrt{3})(\sqrt{7}-\sqrt{3})

Example 36

hard

A square has diagonal

10\sqrt{2}

. Find its side length.

Example 37

challenge

For how many integers

n

with

1\le n\le 500

\sqrt{n}

an integer?

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Related Concepts

Exponents Multiplication Irrational Numbers Pythagorean Theorem

Background Knowledge

These ideas may be useful before you work through the harder examples.

exponentsmultiplication