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.

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.

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.