Question

How do you simplify `2^(5/2) - 2^(3/2)`?

Answer 1

In general
`b^(p+q) = b^p*b^q`
so
`2^(5/2)` can be re-written as `2^(3/2)*2^(2/2) = 2^(3/2)*2`

`2^(5/2)-2^(3/2)`

`= 2^(3/2)(2-1)`

and since
`2^(3/2) = (2^3)^(1/2) = 2sqrt(2)`

`2^(5/2)-2^(3/2) = 2sqrt(2)`

Answer 2

`2sqrt2`

Explanation:

Do not be tempted to simplify the indices. These are separate terms.

However, we can factorise and take out a common factor of `2^(3/2)`.

When you are dividing and the bases are the same, subtract the indices.

`2^(5/2) -2^(3/2) = 2^(3/2)(2^(2/2) -1)`

`=2^(3/2)(2-1) = 2^(3/2)(1)`

`2^(3/2)` can also be written as: `" "2^(2/2+1/2) = 2xx2^(1/2)`

`=2sqrt2`

Another approach to simplify this is:

`2^(3/2) = sqrt(2^3) = sqrt 8 = sqrt(4xx2)`

`=2sqrt2`