Question

How do you evaluate `(56)^(7/2)`?

Answer 1

`351232sqrt(14)" "` as an exact value

`1314189.807" "` to 3 decimal places as an approximate value

Explanation:

First of all lets do this using logs:

Let `x=(56)^(7/2)`

`" "log(x)" "=" "log[ (56)^(7/2)]" " =" " 7/2log(56)~~6.1186...`

`=>color(green)(x=log^(-1)(6.1186..)~~1314189.807)` to 3 decimal places

`color(blue)("This is not a precise solution but will do as a check.")`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider again `x=(56)^(7/2)`

This is the same as: `sqrt(56^7)" "=" "sqrt(56^2xx56^2xx56^2xx56)`

`=56^3sqrt(56)`

Splitting 56 into a product of primes we observe:

Giving:

`56^3sqrt(2^2xx14)`

`2xx56^3sqrt(14)color(green)(~~1314189.807)` to 3 decimal places

`color(blue)("So the exact value is: "351232sqrt(14))`