Fractional indices

The meaning of a^(1/n) and a^(m/n) as roots, converting between surds and indices, and why the base must be positive.

What a fractional index means

The index laws force the meaning: $\left(a^{\frac12}\right)^2 = a^1 = a$, so $a^{\frac12}$ must be $\sqrt{a}$. In the same way, $a^{\frac13}$ is the cube root.

Fractional indices

$$a^{\frac1n} = \sqrt[n]{a} \qquad a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$$

Example 1●

Evaluate $27^{\frac23}$ and $16^{-\frac34}$.

$27^{\frac23} = (\sqrt[3]{27})^2 = 3^2 = 9$

$16^{-\frac34} = \dfrac{1}{(\sqrt[4]{16})^3} = \dfrac{1}{2^3} = \dfrac18$

The negative sign means reciprocal.

Example 2●●

Write $\dfrac{x}{\sqrt{x}}$ and $\dfrac{1}{\sqrt[3]{x^2}}$ as powers of $x$.

$\dfrac{x}{\sqrt{x}} = x^{1 - \frac12} = x^{\frac12}$

$\dfrac{1}{\sqrt[3]{x^2}} = x^{-\frac23}$

Negative and zero bases

Fractional indices are only used with positive bases. For example, $(-4)^{\frac12}$ would be $\sqrt{-4}$, which is not a real number. Also $(-8)^{\frac13} = -2$, yet $(-8)^{\frac26}$ would be $\sqrt[6]{64} = 2$, so the index laws would contradict each other. Irrational indices such as $2^{\sqrt2}$ are defined using closer and closer rational approximations.

Flashcards

$a^{rac1n}$

$\sqrt[n]{a}$

$a^{rac{m}{n}}$

$\left(\sqrt[n]{a}\right)^m$: root first, then power.

$a^{-rac{m}{n}}$

$\dfrac{1}{a^{\frac{m}{n}}}$

Practice

Evaluate $9^{\frac12}$.
Evaluate $8^{\frac13}$.
Evaluate $27^{\frac23}$.
Evaluate $16^{-\frac34}$.
Evaluate $\left(\tfrac49\right)^{-\frac12}$.
Write $\sqrt{x^3}$ as a power of $x$.
Write $\dfrac{1}{\sqrt[3]{x}}$ as a power of $x$.
Which of these is not a real number?
Simplify $x^{\frac12} \times x^{\frac32}$.