Question

How do you simplify `(7^2/3)^(5/2)`?

Answer 1

`( 7 / sqrt (3) )^ 5`

Explanation:

First, let's set out the exponent rules that are relevant:

(1) `(a ^ b)^c = a ^(bc)`

(2) `(e * f ) ^g = e^g * f^g`

(3) `(1 / h) ^ j = h ^-j`

Now we first use rule (2):

`( 7 ^2 / 3) ^( 5/ 2) = (7 ^2 ) ^ ( 5 / 2 ) * (1 / 3) ^( 5 / 2 )`

We can now use rule (3):

`(7 ^2 ) ^ ( 5 / 2 ) * (1 / 3) ^( 5 / 2 ) = (7 ^2 ) ^ ( 5 / 2 ) * 3 ^( - 5 / 2 )`

We use rule (1) to multiply out the left hand term:

`(7 ^2 ) ^ ( 5 / 2 ) * 3 ^( - 5 / 2 ) = 7 ^( 2 * 5 / 2 ) * 3 ^( - 5 / 2 )`

`= 7 ^ ( 5 ) * 3 ^ (- 5 / 2)`

To simplify further we could again use rule (1) and rule (3):

`7 ^ ( 5 ) * 3 ^ (- 5 / 2) = ( 7 / 3 ^ (1 / 2) )^ 5`

As `x^(1/2) -= sqrt (x)`

`( 7 / 3 ^ (1 / 2) )^ 5 =( 7 / sqrt (3) )^ 5`