Fractional indices and surds

Fractional indices as roots, evaluating powers like 16^(3/4), what surds are, and simplifying surds using square factors.

Fractional indices

$$a^{\frac{1}{n}} = \sqrt[n]{a} \qquad a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$$

Example 1●●

Evaluate $16^{\frac{3}{4}}$ and $27^{\frac{2}{3}}$.

$\sqrt[4]{16} = 2$, so $2^3 = 8$.

$\sqrt[3]{27} = 3$, so $3^2 = 9$.

Surds

A surd is a root that is irrational, such as $\sqrt{2}$ or $\sqrt{12}$. To simplify, find the largest square factor: $\sqrt{ab} = \sqrt{a}\sqrt{b}$.

Example 2●●

Simplify $\sqrt{50}$ and $\sqrt{12}$.

$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

Flashcards

$9^{rac{1}{2}}$

$3$

$8^{rac{1}{3}}$

$2$

$sqrt{18}$

$3\sqrt{2}$

Practice

Evaluate $16^{\frac{3}{4}}$.
Evaluate $27^{\frac{2}{3}}$.
Simplify $\sqrt{50}$.
Simplify $\sqrt{12}$.
Evaluate $9^{\frac{1}{2}}$.
Which is a surd?
Simplify $\sqrt{72}$.
Evaluate $4^{-\frac{1}{2}}$.