# Exponents

## Audio-visual

# Numeracy Basics

## Exponents

## Exponents

Exponents, otherwise known as indices or powers, are used as a shorthand notation for repeated multiplication. For example:

5 x 5 x 5 x 5 = 5^{4}

In this case we say that the number 5 is the base and the number 4 is the exponent or power.

Some other examples of exponents, and how they are evaluated, are as follows:

2^{2} = 2 x 2 = 4

2^{3} = 2 x 2 x 2 = 8

2^{4} = 2 x 2 x 2 x 2 = 16

5 x 5 x 5 x 5 =

5^{4}

2^{2} = 2 x 2 = 4

2^{3} = 2 x 2 x 2 = 8

2^{4} = 2 x 2 x 2 x 2 = 16

## Examples: Exponents

1. Evaluate:

a. 3^{2} = 3 x 3

= 9

b. 8^{3} = 8 x 8 x 8

= 512

2. Rewrite the following:

a. 2 x 2 x 2 x 2 = 2^{4}

b. 10 x 10 x 10 x 10 x 10 x 2 x 2 = 10^{5} x 2^{2}

^{4}

^{5}x 2

^{2}

## Activity 1: Practice Questions

## The Exponent Laws

There are ten 'laws' you need to abide by when working with exponents.

When detailing these laws, we will use the variables x and/or y to represent any whole number bases and the variables a and/or b to represent any integer exponents. For example:

x^{a} means x multiplied by itself a times

If x = 3 and a = 4 then x^{a} = 3^{4} = 3 x 3 x 3 x 3

x = 10 and a = 5 then x^{a} = 10^{5} = 10 x 10 x 10 x 10 x 10

x^{a} means x multiplied by itself a times

If x = 3 and a = 4 then x^{a} = 3^{4} = 3 x 3 x 3 x 3

x = 10 and a = 5 then x^{a} = 10^{5} = 10 x 10 x 10 x 10 x 10

## Exponent Laws 1 and 2

Now that we have defined our variables let's look at Exponent Laws 1 and 2, which are the simplest two index laws:

*Law 1: x ^{ 1} = x*

*For example, 5 ^{ 1} = 5*

*Law 2: x ^{ 0} = 1*

*For example, 4 ^{ 0} = 1*

*Law 1: x ^{ 1} = x*

*For example, 5 ^{ 1} = 5*

*Law 2: x ^{ 0} = 1*

*For example, 4 ^{ 0} = 1*

## Exponent Laws 3 and 4

Exponent Laws 3 and 4 cover how to multiply and divide with exponents respectively, and are as follows:

Law 3: x^{a}x^{b} = x^{a+b}

For example, 2^{3}2^{4} = (2 x 2 x 2) x (2 x 2 x 2 x 2)

= 2 x 2 x 2 x 2 x 2 x 2 x 2

= 2^{7}

= 2^{3+4}

Law 4: ^{xa}/_{xb} = x^{a-b}

For example, ^{105}/_{103} = ^{10x10x10x10x10}/_{10x10x10}

= 10 x 10

= 10^{2}

= 10^{5-3}

Law 3: x^{a}x^{b} = x^{a+b}

For example, 2^{3}2^{4} =

(2 x 2 x 2) x (2 x 2 x 2 x 2)

= 2 x 2 x 2 x 2 x 2 x 2 x 2

= 2^{7}

= 2^{3+4}

Law 4: ^{xa}/_{xb} = x^{a-b}

For example, ^{105}/_{103} =

^{10x10x10x10x10}/_{10x10x10}

= 10 x 10

= 10^{2}

= 10^{5-3}

## Examples: Exponent Laws 1 to 4

1.3^{2}3^{3}3^{5} = 3^{2+3+5}

= 310

2.^{5354}/_{56} = ^{53+4}/_{56}

= ^{57}/_{56}

= 5^{7-6}

= 5^{1}

= 5

3.4^{3}4^{-3} = 4^{3+(-3)}

= 4^{3-3}

= 4^{0}

= 1

= 3^{2+3+5}

= 3^{10}

= ^{53+4}/_{56}

= ^{57}/_{56}

= 5^{7-6}

= 5^{1}

= 5

= 4^{3+(-3)}

= 4^{3-3}

= 4^{0}

= 1

## Activity 2: Practice Questions

## Exponent Laws 5 and 6

Exponent Laws 5 and 6 cover negative exponents and exponents of exponents respectively, and are as follows:

*Law 5: x ^{-a}= 1/x^{ a}*

*For example, 2 ^{ -3} = 1/2^{ 3}*

*For example, 4 ^{ -1} = 1/4^{ 1} = 1/4*

*Law 6: (x ^{ a})^{ b} = x^{ ab}*

*For example, (2 ^{ 2})^{ 3} = (2 x 2)^{ 3}*

* = (2 x 2) x (2 x 2) x (2 x 2)*

* = 2 x 2 x 2 x 2 x 2 x 2*

* = 2 ^{ 6}*

* = 2 ^{ 2x3}*

*Law 5: x ^{-a}= 1/x^{ a}*

*For example, 2 ^{ -3} = 1/2^{ 3}*

*For example, 4 ^{ -1} = 1/4^{ 1} = 1/4*

*Law 6: (x ^{ a})^{ b} = x^{ ab}*

*For example, (2 ^{ 2})^{ 3} = *

* (2 x 2) ^{ 3}*

* = (2 x 2) x (2 x 2) x (2 x 2)*

* = 2 x 2 x 2 x 2 x 2 x 2*

* = 2 ^{ 6}*

* = 2 ^{ 2x3}*

## Exponent Laws 7 and 8

Exponent Laws 7 and 8 cover how to simplify when we have two bases and one exponent, and are as follows:

*Law 7: (xy) ^{a}= x ^{a}y ^{a}*

*For example, (4 x 5) ^{3} = (4 x 5) x (4 x 5) x (4 x 5)*

*= 4 x 5 x 4 x 5 x 4 x 5*

*= 4 x 4 x 4 x 5 x 5 x 5*

*= 4 ^{3}5 ^{3}*

*Law 8: (x/y) ^{a}= x ^{a}/y ^{a}*

*For example, (4/5) ^{3}= 4/5 x 4/5 x 4/5*

*= ^{4 x 4 x 4}/_{5 x 5 x 5}*

*= ^{4 3}/_{5 3}*

*Law 7: (xy) ^{a}= x ^{a}y ^{a}*

*For example, (4 x 5) ^{3} =*

*(4 x 5) x (4 x 5) x (4 x 5)*

*= 4 x 5 x 4 x 5 x 4 x 5*

*= 4 x 4 x 4 x 5 x 5 x 5*

*= 4 ^{3}5 ^{3}*

*Law 8: (x/y) ^{a}= x ^{a}/y ^{a}*

*For example, (4/5) ^{3}=*

*4/5 x 4/5 x 4/5*

*= ^{4 x 4 x 4}/_{5 x 5 x 5}*

*= ^{4 3}/_{5 3}*

## Examples: Exponent Laws 5 to 8

1.(5^{-2})^{2} = 5^{-2x2}

= 5^{-4}

= ^{1}/_{54}

2.(3^{-4}4)^{-3} = (3 ^{-4})^{-3}4^{-3}

= 3^{-4x-3}4^{-3}

= 3^{12}4^{-3}

= 3^{12}

= 4^{3}

3.(5/6^{-2})^{3} = 5^{3}/(6^{-2})^{3}

= 5^{3}/6^{-2x3}

= 5^{3}/6^{-6}

= 5^{3}6^{6}

= 5^{-2x2}

= 5^{-4}

= ^{1}/_{54}

= (3 ^{-4})^{-3}4^{-3}

= 3^{-4x-3}4^{-3}

= 3^{12}4^{-3}

= 5^{3}/(6^{-2})^{3}

= 5^{3}/6^{-2x3}

= 5^{3}/6^{-6}

= 5^{3}6^{6}

## Activity 3: Practice Questions

## Exponent Laws 9 and 10

Exponent Laws 9 and 10 cover fractional exponents, and are as follows:

*Law 9: x ^{1/a} = ^{a}√x*

*For example, 9 ^{1/2} = ^{2}√9 = √9 *

*27 ^{1/3} = 3√27 *

*Law 10: x ^{ b/a} = ^{a}√x^{ b} = (^{ a}√x)^{ b}*

*For example, 8 ^{ 2/3} = 8^{ 2 x 1/3} *

*= (8 ^{2}) ^{1/3} (from Law 6)*

*= ^{3}√8 ^{2} (from Law 9)*

*Similarly, 8 ^{2/3} = 8 ^{1/3 x 2}*

*= (8 ^{1/3}) ^{2} (from Law 6)*

*=( ^{3}√8 ) ^{2} (from Law 9)*

*Law 9: x ^{1/a} = ^{a}√x*

*For example, 9 ^{1/2} =*

^{2}√9 =

*√9*

*27 ^{1/3} = *

^{3}√27

*Law 10: x ^{ b/a} = ^{a}√x^{ b}*

*= ( ^{ a}√x)^{ b}*

*For example, 8 ^{ 2/3} =*

*8 ^{ 2 x 1/3} *

*= (8 ^{2}) ^{1/3} (from Law 6)*

*= ^{3}√8 ^{2} (from Law 9)*

*Similarly, 8 ^{2/3} = 8 ^{1/3 x 2}*

*= (8 ^{1/3}) ^{2} (from Law 6)*

*=( ^{3}√8 ) ^{2} (from Law 9)*

## Examples: Exponent Laws 9 and 10

1.16^{1/4} = ^{4}√16

= 2

2.27^{5/3} = (^{3}√27)^{5}

= 3 ^{5}

= 243

= ^{4}√16

= 2

= (^{3}√27)^{5}

= 3 ^{5}

= 243

## Activity 4: Practice Questions

## End of Topic

Congratulations, you have completed this topic.

You should now have a better understanding of Exponents.

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