Roots as Inverse Growth Examples in Math

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

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

Concept 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?'

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: Taking the $n$ th root asks which number, raised to the $n$ th power, rebuilds the value you started with.

Common stuck point: The procedure for roots as inverse growth is the easy part; the trap is treating $\sqrt{a}$ as $a\div 2$ . Asking "Am I given a power's output and asked for the base that produced it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I given a power's output and asked for the base that produced it?

Worked Examples

Example 1

easy

Since

8^2 = 64

, what is

\sqrt{64}

? Explain roots as inverses of powers.

Answer

\sqrt{64} = 8

First step

We know

8^2 = 64

(squaring).

Full solution

2
The square root undoes squaring: $\sqrt{64} = 8$ .
3
Check: $8 \times 8 = 64$ ✓
4
Root is the inverse operation of the corresponding power.

A square root answers: what number times itself gives this result? Since

8^2=64

\sqrt{64}=8

. Roots undo powers.

Example 2

medium

Estimate

\sqrt{50}

to one decimal place by finding the two perfect squares it lies between.

Example 3

medium

Estimate

\sqrt{30}

to one decimal place by bracketing it between perfect squares.

Example 4

medium

Solve

x^2 = 49

for all real

x

Example 5

medium

Solve

\sqrt{x} = 9

for

x

, and verify your answer.

Example 6

hard

Solve

\sqrt{2x + 1} = 5

for

x

, and check for extraneous solutions.

Example 7

hard

Solve

\sqrt{x + 6} = x

for real

x

Example 8

hard

Express

\sqrt[6]{64}

as a power of

2

, then find its value.

Example 9

hard

Simplify

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

Example 10

challenge

Solve

\sqrt{x + 7} - \sqrt{x} = 1

for real

x

Practice Problems

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

Example 1

easy

Find

\sqrt{144}

and

\sqrt[3]{27}

Example 2

medium

Between which two consecutive integers does

\sqrt{75}

lie? Estimate to one decimal.

Example 3

easy

Find

\sqrt{49}

Example 4

easy

Find

\sqrt[3]{27}

Example 5

easy

Find

\sqrt{100}

Example 6

easy

Find

\sqrt[3]{-8}

Example 7

easy

Find

\sqrt{1}

Example 8

easy

\sqrt{16}

equal to

\sqrt[3]{16}

Example 9

easy

Find

\sqrt{0}

Example 10

easy

Find

\sqrt[3]{1000}

Example 11

medium

Simplify

\sqrt{(-5)^2}

Example 12

medium

Solve

x^2 = 36

for all real

x

Example 13

medium

Solve

x^3 = 64

for real

x

Example 14

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 15

medium

Estimate

\sqrt{50}

between two consecutive integers.

Example 16

medium

Find

\sqrt{\frac{9}{16}}

Example 17

medium

\sqrt{x} = 5

, what is

x

Example 18

challenge

For which real numbers

a

does

\sqrt{a^2} = a

hold, and where does it fail? Justify.

Example 19

challenge

Solve

\sqrt{x+4} = x - 2

and check for extraneous solutions.

Example 20

challenge

Without a calculator, explain why

\sqrt[3]{-27} = -3

but

\sqrt{-27}

is not a real number.

Example 21

medium

Simplify

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

Example 22

medium

A square's area grows from

25

100

. By what factor does its side grow?

Example 23

easy

Find

\sqrt{81}

Example 24

easy

Find

\sqrt[3]{64}

Example 25

easy

Find

\sqrt{225}

Example 26

easy

Find

\sqrt[3]{125}

Example 27

easy

Find

\sqrt[4]{16}

Example 28

medium

Simplify

\sqrt{72}

to the form

a\sqrt{b}

with

b

square-free.

Example 29

medium

Solve

x^3 = -27

for real

x

Example 30

medium

Simplify

\sqrt{50} + \sqrt{18}

Example 31

medium

Simplify

\sqrt[3]{54}

to the form

a\sqrt[3]{b}

with

b

cube-free.

Example 32

medium

A square has area

169

^2

. Find its side length.

Example 33

hard

Rationalize and simplify

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

Example 34

hard

Solve

\sqrt{x - 2} + 3 = 7

for

x

Example 35

hard

The volume of a cube is

216

^3

. Find the length of one edge.

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Related Concepts

Square Roots Exponents Radical Operations Exponential Function

Background Knowledge

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

square rootsexponents