# 3.1 Exponents and roots  (Page 2/2)

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We can continue this way to see such roots as fourth roots, fifth roots, sixth roots, and so on.

There is a symbol used to indicate roots of a number. It is called the radical sign $\sqrt[n]{\phantom{\rule{16px}{0ex}}}$

## The radical sign $\sqrt[n]{\phantom{\rule{16px}{0ex}}}$

The symbol $\sqrt[n]{\phantom{\rule{16px}{0ex}}}$ is called a radical sign and indicates the nth root of a number.

We discuss particular roots using the radical sign as follows:

## Square root

$\sqrt[2]{\text{number}}$ indicates the square root of the number under the radical sign. It is customary to drop the 2 in the radical sign when discussing square roots. The symbol $\sqrt[]{\phantom{\rule{16px}{0ex}}}$ is understood to be the square root radical sign.

$\sqrt{\text{49}}$ = 7 since $7\cdot 7={7}^{2}=\text{49}$

## Cube root

$\sqrt[3]{\text{number}}$ indicates the cube root of the number under the radical sign.

$\sqrt[3]{8}=2$ since $2\cdot 2\cdot 2={2}^{3}=8$

## Fourth root

$\sqrt[4]{\text{number}}$ indicates the fourth root of the number under the radical sign.

$\sqrt[4]{\text{81}}=3$ since $3\cdot 3\cdot 3\cdot 3={3}^{4}=\text{81}$

In an expression such as $\sqrt[5]{\text{32}}$

$\sqrt[]{\phantom{\rule{16px}{0ex}}}$ is called the radical sign .

## Index

5 is called the index . (The index describes the indicated root.)

32 is called the radicand .

$\sqrt[5]{\text{32}}$ is called a radical (or radical expression).

## Sample set b

Find each root.

$\sqrt{\text{25}}$ To determine the square root of 25, we ask, "What whole number squared equals 25?" From our experience with multiplication, we know this number to be 5. Thus,

$\sqrt{\text{25}}=5$

Check: $5\cdot 5={5}^{2}=\text{25}$

$\sqrt[5]{\text{32}}$ To determine the fifth root of 32, we ask, "What whole number raised to the fifth power equals 32?" This number is 2.

$\sqrt[5]{\text{32}}=2$

Check: $2\cdot 2\cdot 2\cdot 2\cdot 2={2}^{5}=\text{32}$

## Practice set b

Find the following roots using only a knowledge of multiplication.

$\sqrt{\text{64}}$

8

$\sqrt{\text{100}}$

10

$\sqrt[3]{\text{64}}$

4

$\sqrt[6]{\text{64}}$

2

## Calculators

Calculators with the $\sqrt{x}$ , ${y}^{x}$ , and $1/x$ keys can be used to find or approximate roots.

## Sample set c

Use the calculator to find $\sqrt{\text{121}}$

 Display Reads Type 121 121 Press $\sqrt{x}$ 11

Find $\sqrt[7]{\text{2187}}$ .

 Display Reads Type 2187 2187 Press ${y}^{x}$ 2187 Type 7 7 Press $1/x$ .14285714 Press = 3

$\sqrt[7]{\text{2187}}=3$ (Which means that ${3}^{7}=2187$ .)

## Practice set c

Use a calculator to find the following roots.

$\sqrt[3]{\text{729}}$

9

$\sqrt[4]{\text{8503056}}$

54

$\sqrt{\text{53361}}$

231

$\sqrt[\text{12}]{\text{16777216}}$

4

## Exercises

For the following problems, write the expressions using expo­nential notation.

$4\cdot 4$

${4}^{2}$

$\text{12}\cdot \text{12}$

$9\cdot 9\cdot 9\cdot 9$

${9}^{4}$

$\text{10}\cdot \text{10}\cdot \text{10}\cdot \text{10}\cdot \text{10}\cdot \text{10}$

$\text{826}\cdot \text{826}\cdot \text{826}$

${\text{826}}^{3}$

$3,\text{021}\cdot 3,\text{021}\cdot 3,\text{021}\cdot 3,\text{021}\cdot 3,\text{021}$

$\underset{\text{85 factors of 6}}{\underbrace{6·6·····6}}$

${6}^{\text{85}}$

$\underset{\text{112 factors of 2}}{\underbrace{2·2·····2}}$

$\underset{\text{3,008 factors of 1}}{\underbrace{1·1·····1}}$

${1}^{\text{3008}}$

For the following problems, expand the terms. (Do not find the actual value.)

${5}^{3}$

${7}^{4}$

$7\cdot 7\cdot 7\cdot 7$

${\text{15}}^{2}$

${\text{117}}^{5}$

$\text{117}\cdot \text{117}\cdot \text{117}\cdot \text{117}\cdot \text{117}$

${\text{61}}^{6}$

${\text{30}}^{2}$

$\text{30}\cdot \text{30}$

For the following problems, determine the value of each of the powers. Use a calculator to check each result.

${3}^{2}$

${4}^{2}$

$4\cdot 4=\text{16}$

${1}^{2}$

${\text{10}}^{2}$

$\text{10}\cdot \text{10}=\text{100}$

${\text{11}}^{2}$

${\text{12}}^{2}$

$\text{12}\cdot \text{12}=\text{144}$

${\text{13}}^{2}$

${\text{15}}^{2}$

$\text{15}\cdot \text{15}=\text{225}$

${1}^{4}$

${3}^{4}$

$3\cdot 3\cdot 3\cdot 3=\text{81}$

${7}^{3}$

${\text{10}}^{3}$

$\text{10}\cdot \text{10}\cdot \text{10}=1,\text{000}$

${\text{100}}^{2}$

${8}^{3}$

$8\cdot 8\cdot 8=\text{512}$

${5}^{5}$

${9}^{3}$

$9\cdot 9\cdot 9=\text{729}$

${6}^{2}$

${7}^{1}$

${7}^{1}=7$

${1}^{\text{28}}$

${2}^{7}$

$2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=\text{128}$

${0}^{5}$

${8}^{4}$

$8\cdot 8\cdot 8\cdot 8=4,\text{096}$

${5}^{8}$

${6}^{9}$

$6\cdot 6\cdot 6\cdot 6\cdot 6\cdot 6\cdot 6\cdot 6\cdot 6=\text{10},\text{077},\text{696}$

${\text{25}}^{3}$

${\text{42}}^{2}$

$\text{42}\cdot \text{42}=1,\text{764}$

${\text{31}}^{3}$

${\text{15}}^{5}$

$\text{15}\cdot \text{15}\cdot \text{15}\cdot \text{15}\cdot \text{15}=\text{759},\text{375}$

${2}^{\text{20}}$

${\text{816}}^{2}$

$\text{816}\cdot \text{816}=\text{665},\text{856}$

For the following problems, find the roots (using your knowledge of multiplication). Use a calculator to check each result.

$\sqrt{9}$

$\sqrt{\text{16}}$

4

$\sqrt{\text{36}}$

$\sqrt{\text{64}}$

8

$\sqrt{\text{121}}$

$\sqrt{\text{144}}$

12

$\sqrt{\text{169}}$

$\sqrt{\text{225}}$

15

$\sqrt[3]{\text{27}}$

$\sqrt[5]{\text{32}}$

2

$\sqrt[4]{\text{256}}$

$\sqrt[3]{\text{216}}$

6

$\sqrt[7]{1}$

$\sqrt{\text{400}}$

20

$\sqrt{\text{900}}$

$\sqrt{\text{10},\text{000}}$

100

$\sqrt{\text{324}}$

$\sqrt{3,\text{600}}$

60

For the following problems, use a calculator with the keys $\sqrt{x}$ , ${y}^{x}$ , and $1/x$ to find each of the values.

$\sqrt{\text{676}}$

$\sqrt{1,\text{156}}$

34

$\sqrt{\text{46},\text{225}}$

$\sqrt{\text{17},\text{288},\text{964}}$

4,158

$\sqrt[3]{3,\text{375}}$

$\sqrt[4]{\text{331},\text{776}}$

24

$\sqrt[8]{5,\text{764},\text{801}}$

$\sqrt[\text{12}]{\text{16},\text{777},\text{216}}$

4

$\sqrt[8]{\text{16},\text{777},\text{216}}$

$\sqrt[\text{10}]{9,\text{765},\text{625}}$

5

$\sqrt[4]{\text{160},\text{000}}$

$\sqrt[3]{\text{531},\text{441}}$

81

## Exercises for review

( [link] ) Use the numbers 3, 8, and 9 to illustrate the associative property of addition.

( [link] ) In the multiplication $8\cdot 4=\text{32}$ , specify the name given to the num­bers 8 and 4.

8 is the multiplier; 4 is the multiplicand

( [link] ) Does the quotient $\text{15}÷0$ exist? If so, what is it?

( [link] ) Does the quotient $0÷\text{15}$ exist? If so, what is it?

Yes; 0

( [link] ) Use the numbers 4 and 7 to illustrate the commutative property of multiplication.

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