Question

How do you simplify `(a^4)^-5 * a^13`?

Answer 1

By definition, `x^(-N)=1/(x^N)`.
Therefore,
`(a^4)^(-5)=1/((a^4)^5)`

By definition. `(x^M)^N = (x^M) * (x^M) * (x^M)...`
(where multiplication is performed `N` times)
But each `x^M = x*x*x...`
(where multiplication is performed `M` times)
Therefore,
`(x^M)^N = x*x*x*...`
(where multiplication is performed `M*N` times).
Hence, `(x^M)^N=x^(M*N)`

Using the above, we can write:
`(a^4)^(-5)=1/((a^4)^5)=1/a^(4*5)=1/a^20`

The original expression, therefore, equals to
`(a^4)^(-5)*a^13=a^13/a^20=1/a^7=a^(-7)`