How do you evaluate fractional exponents?

1 Answer
Dec 20, 2014

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)