# Lesson 6: Negative (and zero) indices

Introduction

By the end of this lesson, you should understand what negative (and zero) indices represent and how to work with them. You'll need to be able to add, subtract and multiply with negative numbers, though. If you're not confident doing this, go back and revise it before you attempt this lesson.

SECTION A

$$a^3~=~\Large\frac{a^4}{a^1}$$

$$a^2~=~\Large\frac{a^3}{a^1}$$

$$a^1~=~\Large\frac{a^2}{a^1}\normalsize~~=~a$$

Dividing by  $$a$$  each time causes the index to go down by 1. But it doesn't stop here ...

$$a^0~=~\Large\frac{a^1}{a^1}\normalsize~~=~1$$

$$a^{-1}~=~\Large\frac{1}{a^1}$$

$$a^{-2}~=~\Large\frac{1}{a^2}$$

$$a^{-3}~=~\Large\frac{1}{a^3}$$

Continuing the pattern allows us to see where zero and negative indices come from and what they mean.

Zero index

Anything raised to the power of 0 is just 1. Yep, anything and always!

Negative indices

You need to be able to work with the form  $$x^{-n}$$  but write your answer in the form  $$\Large\frac{1}{x^n}$$ .

Examples

1. Simplify  $$m^0$$

2. Simplify  $$w^{-3}$$

Answer:  $$\Large\frac{1}{w^3}$$

# Practise to master

SECTION A

Simplify (write in correct form) each of these using your knowledge of indices.

01)  $$c^{-2}$$

$$\Large\frac{1}{c^2}$$

02)  $$x^0$$

$$1$$

03)  $$a^{-7}$$

$$\Large\frac{1}{a^7}$$

04)  $$y^0$$

$$1$$

05)  $$u^{-5}$$

$$\Large\frac{1}{u^5}$$

06)  $$g^{-1}$$

$$\Large\frac{1}{g^1}\normalsize~=~\Large\frac{1}{g}$$

SECTION B

This section mixes in zero and negative indices with everything else you've learned so far. Don't forget, you must be able to add, subtract and multiply with negative numbers to complete this successfully.

Example 1

Simplify  $$2a^3\times 4a^{-5}$$

Answer:  $$8a^{3+-5}~=~8a^{3-5}~=~8a^{-2}~=~\Large\frac{8}{a^2}$$

As you know, when multiplying terms, we add powers  $$(3+-5=3-5=-2)$$

Example 2

Simplify  $$(3a^{-2})^{-3}$$

Answer: $$(3)^{-3}(a^{-2})^{-3}~=~3^{-3}a^{-2\times -3}~=~3^{-3}a^6~=~\Large\frac{a^6}{3^3}\normalsize~=~\Large\frac{a^6}{27}$$

When raising terms (with indices) to a power, we multiply powers  $$(-2\times -3=6)$$

Example 3

Simplify  $$a^{-2}\div a^{-4}$$

Answer: $$a^{-2--4}~=~a^{-2+4}~=~a^2$$

When dividing terms, we subtract powers, of course  $$(-2--4=-2+4=2)$$

Example 4

Simplify  $$2x^3yz\div 4x^2y^5z$$

Answer: $$\Large\frac{1}{2}\normalsize~~x^{3-2}y^{1-5}z^{1-1}~=~\Large\frac{1}{2}\normalsize~~x^{1}y^{-4}z^0~=~\Large\frac{x}{2y^4}$$

Could you explain each step of this to someone who didn't understand it?

# Practise to master

SECTION B

Simplify these expressions using your knowledge of indices.

01)  $$x^{-3}\times x^{-5}$$

$$x^{-3+-5}~=~x^{-3-5}~=~x^{-8}~=~\Large\frac{1}{x^8}$$

02)  $$(b^{-3})^{-2}$$

$$b^{-3\times -2}~=~b^6$$

03)  $$x^{-2}\div x^{-5}$$

$$x^{-2--5}~=~x^{-2+5}~=~x^3$$

04)  $$\Large\frac{v^5}{v^{-2}}$$

$$v^{5--2}~=~v^{5+2}~=~v^7$$

05)  $$y^3\times y^{-4}$$

$$y^{3+-4}~=~y^{3-4}~=~y^{-1}~=~\Large\frac{1}{y^1}\normalsize~=~\Large\frac{1}{y}$$

06)  $$(r^{-2})^4$$

$$r^{-2\times 4}~=~r^{-8}~=~\Large\frac{1}{r^8}$$

07)  $$y^{-4}\div y^2$$

$$y^{-4-2}~=~y^{-6}~=~\Large\frac{1}{y^6}$$

08)  $$\Large\frac{x^{-3}}{x^2}$$

$$x^{-3-2}~=~x^{-5}~=~\Large\frac{1}{x^5}$$

09)  $$e^{-2}\times e\times e^3$$

$$e^{-2+1+3}~=~e^2$$

10)  $$(2g^2)^{-5}$$

$$2^{-5}(g^2)^{-5}~=~2^{-5}g^{2\times -5}~=~2^{-5}g^{-10}~=~\Large\frac{1}{2^5g^{10}}\normalsize~=~\Large\frac{1}{32g^{10}}$$

11)  $$6a^{-5}\div 3a^3$$

$$2a^{-5-3}~=~2a^{-8}~=~\Large\frac{2}{a^8}$$

12)  $$\Large\frac{5e^3}{e^5}$$

$$5e^{3-5}~=~5e^{-2}~=~\Large\frac{5}{e^2}$$

13)  $$2w^{-3}\times 3w^{-2}$$

$$6w^{-3+-2}~=~6w^{-3-2}~=~6w^{-5}~=~\Large\frac{6}{w^5}$$

14)  $$(4u^{-2})^2$$

$$4^2\times (u^{-2})^2~=~16u^{-2\times 2}~=~16u^{-4}~=~\Large\frac{16}{u^4}$$

15)  $$2x^6\div 6x^{-2}$$

$$\Large\frac{2x^6}{6x^{-2}}\normalsize~=~\Large\frac{1x^{6--2}}{3}\normalsize~=~\Large\frac{x^{6+2}}{3}\normalsize~=~\Large\frac{x^{8}}{3}$$

16)  $$\Large\frac{8s^3}{2s^7}$$

$$4s^{3-7}~=~4s^{-4}~=~\Large\frac{4}{s^4}$$

17)  $$3m^{-4}n^3\times 4m^2n^{-7}$$

$$12m^{-4+2}n^{3+-7}~=~12m^{-2}n^{3-7}~=~12m^{-2}n^{-4}~=~\Large\frac{12}{m^2n^4}$$

18)  $$(3xy^3)^{-2}$$

$$3^{-2}x^{-2}(y^3)^{-2}~=~3^{-2}x^{-2}y^{3\times -2}~=~3^{-2}x^{-2}y^{-6}~=~\Large\frac{1}{3^2x^2y^6}\normalsize~=~\Large\frac{1}{9x^2y^6}$$

19)  $$9x^{-5}y^9\div 3x^2y^{-2}$$

$$3x^{-5-2}y^{9--2}~=~3x^{-7}y^{9+2}~=~3x^{-7}y^{11}~=~\Large\frac{3y^{11}}{x^7}$$

20)  $$\Large\frac{18xy^2}{8x^3y^9}$$

$$\Large\frac{9x^{1-3}y^{2-9}}{4}\normalsize~=~\Large\frac{9x^{-2}y^{-7}}{4}\normalsize~=~\Large\frac{9}{4x^2y^7}$$

21)  $$5p^2q^{-3}\times 2p^{-6}q^{-2}$$

$$10p^{2+-6}q^{-3+-2}~=~10p^{2-6}q^{-3-2}~=~10p^{-4}q^{-5}~=~\Large\frac{10}{p^4q^5}$$

22)  $$(2a^3b^{-2})^4$$

$$2^4(a^3)^4(b^{-2})^4~=~16a^{3\times 4}b^{-2\times 4}~=~16a^{12}b^{-8}~=~\Large\frac{16a^{12}}{b^8}$$

23)  $$4x^5y^{-8}\div 8x^{-2}y^{-7}$$

$$\Large\frac{4x^5y^{-8}}{8x^{-2}y^{-7}}\normalsize~=~\Large\frac{1x^{5--2}y^{-8--7}}{2}\normalsize~=~\Large\frac{1x^{5+2}y^{-8+7}}{2}\normalsize~=~\Large\frac{x^7y^{-1}}{2}\normalsize~=~\Large\frac{x^7}{2y}$$

24)  $$\Large\frac{4m^7n^3}{10m^6n^5}$$

$$\Large\frac{2m^{7-6}n^{3-5}}{5}\normalsize~=~\Large\frac{2m^1n^{-2}}{5}\normalsize~=~\Large\frac{2m}{5n^2}$$