Pre-Algebra

Square Roots

Lesson

A square root undoes a square. If $a \cdot a = n$ , then $\sqrt{n} = a$ .

Read $\sqrt{25}$ as “the square root of 25”, and the answer is $5$ because $5 \cdot 5 = 25$ .

Perfect squares to memorize

These come up everywhere:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169

Their square roots are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13$ .

Worked example 1

\sqrt{49} = 7

Because $7 \cdot 7 = 49$ .

Worked example 2 — area to side

A square has area $81$ square ft. Its side length is:

\sqrt{81} = 9 \text{ ft}

Estimation

For numbers that aren’t perfect squares, sandwich them between two perfect squares. $\sqrt{20}$ sits between $\sqrt{16} = 4$ and $\sqrt{25} = 5$ , so it’s between $4$ and $5$ .

Note: by convention, $\sqrt{n}$ means the positive root. So $\sqrt{25} = 5$ , not $\pm 5$ .

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\sqrt{16}

Problem 2

\sqrt{25}

Problem 3

\sqrt{81}

Problem 4

\sqrt{100}

Practice

Standard problems matching the lesson.

Problem 5

\sqrt{1}

Problem 6

\sqrt{36}

Problem 7

\sqrt{49}

Problem 8

\sqrt{64}

Problem 9

\sqrt{144}

Problem 10

\sqrt{121}

Problem 11

\sqrt{169}

Problem 12

\sqrt{0}

Problem 13

\sqrt{9} + \sqrt{16}

Problem 14

\sqrt{36} - \sqrt{4}

Problem 15

\text{Between which two integers is } \sqrt{20}? \text{ Enter the smaller.}

Problem 16

\text{Between which two integers is } \sqrt{50}? \text{ Enter the smaller.}

Problem 17

\sqrt{81} - \sqrt{16}

Problem 18

\sqrt{100} + \sqrt{9}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

\sqrt{225}

Problem 20

\sqrt{4} \cdot \sqrt{25}

Problem 21

\sqrt{100} - \sqrt{36}

Problem 22

\sqrt{49} + \sqrt{81}

Problem 23

\text{Between which two integers is } \sqrt{75}? \text{ Enter the smaller.}

Problem 24

\sqrt{196}

Problem 25

4 \cdot \sqrt{169}

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