Question

How do you simplify `(8^ { 1/ 3} x ^ { 2/ 5} ) ^ { 1/ 2}`?

Answer 1

`8^(1/6)x^(1/5)`

`8^(1/6) =root(6)(8)~~1.4142134....`

Note that 1.4142134 is not as precise an answer as `8^(1/6)`

Explanation:

There are two ways of writing square root.
Suppose we wished to square root the value of `a`
Note that `a` is the same thing as `a^1`

Then we could write it as: `sqrt(a^1)`
Or we could write it as:`" "a^(1xx1/2) = a^(1/2)`
They both mean the same thing.

Suppose you has `sqrt(a^3)`
This is the same thing as `a^(3xx1/2) = a^(2/3)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
You are taking the squire root of everything inside the brackets.

`(8^(1/3)x^(2/5))^(1/2)" "->" "8^(1/3xx1/2)xxx^(2/5xx1/2)`
`" "=8^(1/6)x^(2/10)`

`" "=8^(1/6)x^(1/5)`

Answer 2

`8^(1/6)x^(1/5) =root6(8) root5(x)`

While this can be calculated, the answer is an irrational number.

Explanation:

Use the power law of indices to remove the bracket.

`(x^m)^n = x^(mxxn)`

`=8^(1/3xx1/2)x^(2/5xx1/2)`

`=8^(1/6)x^(1/5)`

This can also be written as `root6(8) root5(x)`