Question

How do you write `625^(3/4)` in radical form?

Answer 1

Notice that `625=5^4` hence

`root 4 (625^3)=root 4 (5^12)=5^3=125`

Answer 2

If you don't automatically know that `5^4 = 625`, or that `25^2 = 625`, another way to do this is:

`color(blue)(625^"3/4")`

`= (600 + 25)^"3/4"`

`= (60*10 + 25)^"3/4"`

`= (120*5 + 25)^"3/4"`

`= (12*50 + 25)^"3/4"`

`= (24*25 + 25)^"3/4"`

`= (25^2)^"3/4"`

Remember that `(x^a)^b = x^(a*b)`.

`= 25^(2*"3/4")`

`= 25^("6/4")`

`= 25^("3/2")`

`= 25^(3*"1/2")`

`= (5^2)^(3*"1/2")`

`= 5^(2*3*"1/2")`

Remember that multiplication is commutative.

`= 5^(3*2*"1/2")`

`= (5*5*5)^(2*"1/2")`

`= (25*5)^(1)`

`= color(blue)(125)`