Question

How do you simplify `a^-2times a^3timesa^4`?

Answer 1

`a^-2 xx a^3 xx a^4 = a^5`

Explanation:

If `n` is a positive integer then:

`a^n = overbrace(a xx a xx .. xx a)^"n times"`

So if `m` and `n` are both positive integers:

`a^m * a^n = overbrace(a xx a xx .. xx a)^"m times" xx overbrace(a xx a xx .. xx a)^"n times"`

`= overbrace(a xx a xx .. xx a)^"m + n times" = a^(m+n)`

If `a != 0` then we can also define:

`a^(-n) = 1/underbrace(a xx a xx .. xx a)_"n times"`

In fact, if `a != 0` then `a^m * a^n = a^(m+n)` regardless of whether `m` and `n` are positive, negative or zero.

`color(white)()`
In our example,

`a^-2 xx a^3 xx a^4 = a^(-2+3+4) = a^5`

Or if you prefer to see it a little slower:

`a^-2 xx a^3 xx a^4`

`=1/(a xx a) xx (a xx a xx a) xx (a xx a xx a xx a)`

`=(color(red)(cancel(color(black)(a xx a))) xx a xx a xx a xx a xx a)/color(red)(cancel(color(black)((a xx a))))`

`=a xx a xx a xx a xx a = a^5`