Question

How do you evaluate fractional exponents?

Answer 1

`x^(a/b) =rootb(x^a) = (rootb(x))^a`

You can just remember this rule, or you can learn about why this is:

fractional exponent `1/b`

So first we're going to look at an expression of the form: `x^(1/b)`.
To investigate what this means, we need to go from `x to x^(1/b)` and then deduce something from it.

`x^1 = x^(b/b) = x^(1/b*b)`
What does multiplication mean? Repeated addition. So we can instead of multiplying by b, adding the number to itself `b` times.
`x^(1/b+1/b+1/b+1/b +...)` (b times)

There is a rule you use when multiplying numbers with the same radical: add the exponents. If we reverse this rule, we get:
`x^(1/b)*x^(1/b)*x^(1/b)*x^(1/b)*x^(1/b)...` (b times)

Now, we still know that this number is equal to `x`. So now we have to think a bit. What number, multiplied by itself b times, gives you `x`.
It's the bth-root of `x` => `x^(1/b)=rootbx`

For example: `8^(1/3)`
If we multiply this by itself 3 times we get:
`8^(1/3)*8^(1/3)*8^(1/3) = 8^(3/3) = 8`
What number multiplied by itself 3 times, gives you 8.
It's of course `root3(8) = 2`

What about `a/b`?
To know what `x^(a/b)` means, we can further rely on our previous findings:
`x^(a/b) = x^(a*1/b) = x^(1/b+1/b+1/b+1/b...)` (a times)
`= x^(1/b)*x^(1/b)*x^(1/b)...` (a times)

Repeated multiplication is equal to exponentiation, so we can write:
`= (x^(1/b))^a = (rootbx)^a`

You can also bring the exponent in the root:
`= rootb(x^a)`