Lesson 5: Dividing terms (with indices)


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!


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...

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


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


Simplify each of these expressions using your knowledge of indices.

01)  \(a^6\div a^2\)


02)  \(\Large\frac{m^5}{m^4}\)


03)  \(v^4\div v^2\)


04)  \(\Large\frac{c^5}{c}\)


05)  \(y^9\div y^5\)


06)  \(\Large\frac{x^8}{x^3}\)


07)  \(w^7\div w\)


08)  \(\Large\frac{n^6}{n^5}\)



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


Simplify these expressions using your knowledge of indices.

01)  \(8b^6\div 2b^2\)


02)  \(6y^7\div 4y^3\)


03)  \(\Large\frac{9e^5}{3e^2}\)


04)  \(\Large\frac{14x^4}{6x}\)


05)  \(4x^6y^2\div 2x^3y\)


06)  \(20a^9b^2\div 12a^7b\)


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


08)  \(\Large\frac{18m^5n^4}{4mn^3}\)


09)  \(16u^2v^7\div uv\)


10)  \(8x^5y^9\div 6x^2y^8\)


11)  \(\Large\frac{10g^5h^4}{5g^3h}\)


12)  \(\Large\frac{7x^5y^6}{6xy^5}\)