Question

How do you evaluate `32` to the power of `2/5`? `32^(2/5)`

Answer 1

`4`
There is a choice of methods.. Work smarter, not harder!

Explanation:

One of the laws of indices deals with cases where there are powers and roots at the same time.

`x^(p/q) = rootq(x^p) = (rootq x)^p`

The denominator shows the root and the numerator gives the power.

Note that the power can be inside or outside the root.

I prefer to find the root first, and then raise to the power because this keeps the numbers smaller. They can usually be calculated mentally rather than needing a calculator

`32^(2/5) = (color(red)root5(32))^2`

=`color(red)(2)^2color(white)(wwwwwwwwwwwww)(2*2*2*2*2=2^5=32)`

=`4`

Compare this with the other method of squaring first.

`root5(color(blue)(32^2)) = root5(color(blue)(1024))`

=`4`

While I know that `2^5 = 32`, the square of `32` and the fifth root of `1024` are not facts that I would be able to recall from memory.