# Lesson 5: Dividing terms (with indices)

Introduction

With multiplying terms, you had a lesson on 'the basics' before you had to deal with indices. With division, there aren't really any basics - it's pretty much all about indices.

There is one basic thing - how to write division. You'll see below that I've used a  $$\div$$  sign. Not the best way to write division in algebra! But in my defense, it helps me with my explanation! And you might get a question written like this in an exam so you do need to be familiar with it. The proper way to write division, though, is as a fraction!

SECTION A

So, you're going to learn how to simplify something like  $$a^5\div a$$ .

If we open it up, we get  $$a\times a\times a\times a\times a\div a$$

Focus on the last two terms. If you multiply by  $$a$$  then divide by  $$a$$ , you end up back where you started ( $$3\times 5\div 5~=~3$$ ). The last two operations cancel each other out!

$$\require{cancel} a\times a\times a \times a \times \cancel{a}\div \cancel{a}~=~a\times a\times a\times a~=~a^4$$

We call this cancelling terms.

Now we'll try  $$a^5\div a^2$$  but let's write this one as a fraction.

$$\Large\frac{a^5}{a^2}\normalsize~=~\Large\frac{a\times a\times a\times \cancel{a}\times \cancel{a}}{\cancel{a}\times \cancel{a}}\normalsize~=~a\times a\times a~=~a^3$$

You just did 5 take away 2, right? I feel a formal rule coming on...

$$x^a\div x^b~=~\Large\frac{x^a}{x^b}\normalsize~=~x^{a-b}$$

Examples

1. Simplify  $$x^7\div x^2$$

Answer:  $$x^{7-2}~=~x^5$$

2. Simplify  $$\Large\frac{b^3}{b}$$

Answer: $$b^{3-1}~=~b^2$$      Please don't forget the invisible power of 1!

# Practise to master

SECTION A

Simplify each of these expressions using your knowledge of indices.

01)  $$a^6\div a^2$$

$$a^{6-2}~=~a^4$$

02)  $$\Large\frac{m^5}{m^4}$$

$$m^{5-4}~=~m^1~=~m$$

03)  $$v^4\div v^2$$

$$v^{4-2}~=~v^2$$

04)  $$\Large\frac{c^5}{c}$$

$$c^{5-1}~=~c^4$$

05)  $$y^9\div y^5$$

$$y^{9-5}~=~y^4$$

06)  $$\Large\frac{x^8}{x^3}$$

$$x^{8-3}~=~x^5$$

07)  $$w^7\div w$$

$$w^{7-1}~=~w^6$$

08)  $$\Large\frac{n^6}{n^5}$$

$$n^{6-5}~=~n^1~=~n$$

SECTION B

If there's more than one term on the top and/or bottom, just deal with them one matching pair at a time. Start by simplifying any numbers the way you'd simplify a 'normal' fraction.

Example 1

Simplify  $$\Large\frac{6x^5}{2x^2}$$

Answer:  $$3x^{5-2}~=~3x^3$$

Since 6 divided by 2 is exactly 3, the denominator (the bottom bit) disappears.

Example 2

Simplify  $$12x^7y^3\div 8x^2y^2$$

Answer: $$\Large\frac{12x^7y^3}{8x^2y^2}\normalsize~~=~\Large\frac{3x^{7-2}y^{3-2}}{2}\normalsize~~=~\Large\frac{3x^5y}{2}$$

Since 8 does NOT divide exactly into 12, the denominator does not disappear and the answer is written as a fraction. So I thought I might as well switch to fraction form straight away, rather than use the  $$\div$$  sign.

# Practise to master

SECTION B

Simplify these expressions using your knowledge of indices.

01)  $$8b^6\div 2b^2$$

$$4b^{6-2}~=~4b^4$$

02)  $$6y^7\div 4y^3$$

$$\Large\frac{6y^7}{4y^3}\normalsize~=~\Large\frac{3y^{7-3}}{2}\normalsize~=~\Large\frac{3y^4}{2}$$

03)  $$\Large\frac{9e^5}{3e^2}$$

$$3e^{5-2}~=~3e^3$$

04)  $$\Large\frac{14x^4}{6x}$$

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

05)  $$4x^6y^2\div 2x^3y$$

$$2x^{6-3}y^{2-1}~=~2x^3y$$

06)  $$20a^9b^2\div 12a^7b$$

$$\Large\frac{20a^9b^2}{12a^7b}\normalsize~=~\Large\frac{5a^{9-7}b^{2-1}}{3}\normalsize~=~\Large\frac{5a^2b}{3}$$

07)  $$\Large\frac{12x^7y^4}{6x^6y^2}$$

$$2x^{7-6}y^{4-2}~=~2xy^2$$

08)  $$\Large\frac{18m^5n^4}{4mn^3}$$

$$\Large\frac{9m^{5-1}n^{4-3}}{2}\normalsize~=~\Large\frac{9m^4n}{2}$$

09)  $$16u^2v^7\div uv$$

$$16u^{2-1}v^{7-1}~=~16uv^6$$

10)  $$8x^5y^9\div 6x^2y^8$$

$$\Large\frac{8x^5y^9}{6x^2y^8}\normalsize~=~\Large\frac{4x^{5-2}y^{9-8}}{3}\normalsize~=~\Large\frac{4x^3y}{3}$$

11)  $$\Large\frac{10g^5h^4}{5g^3h}$$

$$2g^{5-3}h^{4-1}~=~2g^2h^3$$

12)  $$\Large\frac{7x^5y^6}{6xy^5}$$

$$\Large\frac{7x^{5-1}y^{6-5}}{6}\normalsize~=~\Large\frac{7x^4y}{6}$$