This topic will cover how to:

• Evaluate exponents and write repeated multiplication using exponents; and
• Apply the ten exponent laws to rearrange and simplify 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 = 5⁴

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:

2² = 2 x 2 = 4

2³ = 2 x 2 x 2 = 8

2⁴ = 2 x 2 x 2 x 2 = 16

1. Evaluate:

a. 3² = 3 x 3

= 9

b. 8³ = 8 x 8 x 8

= 512

2. Rewrite the following:

a. 2 x 2 x 2 x 2 = 2⁴

b. 10 x 10 x 10 x 10 x 10 x 2 x 2 = 10⁵ x 2²

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 = 3⁴ = 3 x 3 x 3 x 3

x = 10 and a = 5 then x^a = 10⁵ = 10 x 10 x 10 x 10 x 10

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

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

Law 3: x^ax^b = x^a+b

For example, 2³2⁴ = (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

= 2⁷

= 2³⁺⁴

Law 4: ^xa/_x^b = x^a-b

For example, ¹⁰⁵/_10³ = ^{10x10x10x10x10}/_10x10x10

= 10 x 10

= 10²

= 10^5-3

1.3²3³3⁵ = 3²⁺³⁺⁵

= 310

2.⁵³⁵⁴/_5⁶ = ⁵³⁺⁴/_5⁶

= ⁵⁷/_5⁶

= 5^7-6

= 5¹

= 5

3.4³4^-3 = 4^3+(-3)

= 4^3-3

= 4⁰

= 1

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

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 ^ay ^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 ³5 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}

1.(5^-2)² = 5^-2x2

= 5^-4

= ¹/_5⁴

2.(3^-44)^-3 = (3 ^-4)^-34^-3

= 3^-4x-34^-3

= 3¹²4^-3

= 3¹²

= 4³

3.(5/6^-2)³ = 5³/(6^-2)³

= 5³/6^-2x3

= 5³/6^-6

= 5³6⁶

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)

1.16^1/4 = ⁴√16

= 2

2.27^5/3 = (³√27)⁵

= 3 ⁵

= 243

Congratulations, you have completed this topic.

You should now have a better understanding of Exponents.