You know the exponent rules for whole-number exponents, and you know cube roots. What could an exponent like $\frac{1}{2}$ or $\frac{2}{3}$ mean? We get to choose, and the wise choice is the one that keeps the old rules working.
Why it works
The power-of-a-power rule says $(x^a)^b = x^{ab}$. If that is to hold for $a = \frac{1}{2}$, then $\left(x^{1/2}\right)^2 = x^{1} = x$. So $x^{1/2}$ must be a number whose square is $x$: that is $\sqrt{x}$. In the same way, $\left(x^{1/3}\right)^3 = x$, so $x^{1/3} = \sqrt[3]{x}$. And $x^{2/3} = \left(x^{1/3}\right)^2$: take the cube root, then square. Every other meaning would break a rule we already trust.
| Exponent | $8^{\text{exponent}}$ | How |
|---|---|---|
| $\frac{1}{3}$ | $2$ | $\sqrt[3]{8} = 2$ |
| $\frac{2}{3}$ | $4$ | $\left(\sqrt[3]{8}\right)^2 = 2^2$ |
| $1$ | $8$ | $2^3$ |
| $\frac{4}{3}$ | $16$ | $2^4$ |
| $-\frac{1}{3}$ | $\frac{1}{2}$ | $\frac{1}{\sqrt[3]{8}}$ |
Key idea
$x^{1/n} = \sqrt[n]{x}$ and $x^{m/n} = \left(\sqrt[n]{x}\right)^m = \sqrt[n]{x^m}$. The denominator is the root, the numerator is the power. A negative exponent still means a reciprocal.
Worked examples
Worked example: Root, then power
Evaluate $16^{3/4}$.
- Denominator $4$: the fourth root. $\sqrt[4]{16} = 2$, since $2^4 = 16$.
- Numerator $3$: cube it. $2^3 = 8$.
Answer: $16^{3/4} = 8$
Worked example: A negative exponent
Evaluate $27^{-2/3}$.
- $\sqrt[3]{27} = 3$, and $3^2 = 9$, so $27^{2/3} = 9$.
- The negative exponent means the reciprocal: $27^{-2/3} = \frac{1}{9}$.
Answer: $\frac{1}{9}$
Worked example: Change forms
Write $\sqrt[5]{x^3}$ as a power of $x$.
- The root index $5$ goes in the denominator, the power $3$ in the numerator.
Answer: $x^{3/5}$
Common mistake
Computing $8^{1/3}$ as $\frac{8}{3}$, or $16^{3/4}$ as $16 \cdot \frac{3}{4} = 12$. An exponent is never multiplied into the base. Check by undoing: if $8^{1/3}$ were $\frac{8}{3}$, then cubing it would give back $8$, but $\left(\frac{8}{3}\right)^3 \approx 19$.