** Fractional exponents** - this is how it was explained, basically

You know the property a^{m} * a^{n} = a^{m+n}

So a^{1/2} * a^{1/2} = a^{1/2+1/2} = a^{1} =
a

Since the square root of a is a number multiplied by itself to give a, it is
shown that a^{1/2} = √a

√a can also be written as ^{2}√a, which
can also be written ^{2}√a^{1}

If you notice, there is both a 1 and a 2 there, as in 1/2.

a^{1/3} * a^{1/3} * a^{1/3}
= a^{1}

a^{1/3} is multiplied by itself 3 times
to get a, so it must be the cube root of a, which is written ^{3}√a

The rule is: for fractional exponents, the
denominator is the root (e.g. square root = 2, cube root = 3, etc.) and the
numerator is the power (e.g. √a^{1})

So, 5^{3/4} = ^{4}√5^{3}
= ^{4}√125

16^{3/2} = ^{2}√16^{3}
--by the way, ^{2}√(16^{3}) = (^{2}√16)^{3}--

^{2}√16 = 4, 4^{3} = **64**