Practice Roots as Inverse Growth in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Roots reverse the process of exponentiation: the $n$ th root of $a$ finds the number that, raised to the $n$ th power, produces $a$ . For example, $\sqrt[3]{8} = 2$ because $2^3 = 8$ .

If $3^2 = 9$ , then $\sqrt{9} = 3$ . The root asks: 'What number squared gives 9?'

Showing a random 20 of 50 problems.

Example 1

easy

Find

\sqrt{100}

Example 2

easy

Find

\sqrt{144}

and

\sqrt[3]{27}

Example 3

challenge

For which real numbers

a

does

\sqrt{a^2} = a

hold, and where does it fail? Justify.

Example 4

medium

Estimate

\sqrt{30}

to one decimal place by bracketing it between perfect squares.

Example 5

easy

True or false:

\sqrt{36} = -6

Example 6

medium

Solve

\sqrt{x} = 9

for

x

, and verify your answer.

Example 7

medium

A square has area

A

and you find its side is

9

. What was

A

, and what operation recovers the side from

A

Example 8

easy

\sqrt{16}

equal to

\sqrt[3]{16}

Example 9

easy

Find

\sqrt[3]{-8}

Example 10

easy

Find

\sqrt[3]{27}

Example 11

medium

Simplify

\sqrt{36} + \sqrt[3]{8}

Example 12

medium

Between which two consecutive integers does

\sqrt{75}

lie? Estimate to one decimal.

Example 13

hard

Simplify

\sqrt{12} \cdot \sqrt{27}

Example 14

hard

Rationalize and simplify

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

Example 15

easy

Find

\sqrt[3]{125}

Example 16

medium

Solve

x^3 = 64

for real

x

Example 17

medium

A square's area grows from

25

100

. By what factor does its side grow?

Example 18

medium

Solve

x^2 = 49

for all real

x

Example 19

easy

Since

8^2 = 64

, what is

\sqrt{64}

? Explain roots as inverses of powers.

Example 20

medium

Which is larger:

\sqrt{40}

2\sqrt{10}

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Related Concepts

Square Roots Exponents Radical Operations Exponential Function