Fractional exponents

x^(1/2) is √x. Fractions in the exponent are roots in disguise.

A fractional exponent is a root in disguise.

Try this

p in 8^(p/3)

8^(p/3) = (∛8)ᵖ = 2ᵖ = 8^(1/3) = 2

Reading a fractional exponent

x^(1/n) = ⁿ√x — the denominator is the root.
x^(p/q) = ( q√x )ᵖ — root by the denominator, power by the numerator (either order works).
Example: 8^(1/3) = ∛8 = 2, so 8^(2/3) = 2² = 4.

Your turn

Evaluate 27^(2/3).

A fractional exponent is just a tidy way to write a root, so all the normal exponent laws still apply to it.

Rule

x^{(} 1/ n) =^{n} x

Try it

8^(1/3)

Cube root of 8 = 2.

Try it

16^(3/4)

(16^(1/4))³ = 2³ = 8.