Indices & Surds
Year 9 🔢 Number Laws of indices, simplify surds, rationalise denominators.
⚡ Laws of Indices
The index laws let you simplify expressions with powers quickly and accurately.
⚡ Index Laws
$$a^m \times a^n = a^{m+n}$$$$a^m \div a^n = a^{m-n}$$$$(a^m)^n = a^{mn}$$$$a^0 = 1 \qquad a^{-n} = \frac{1}{a^n}$$$$a^{\frac{1}{n}} = \sqrt[n]{a} \qquad a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$$Example: $27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9$
√ Introduction to Surds
A surd is a square root (or other root) that gives an irrational result — it cannot be simplified to a fraction.
Surds: $\sqrt{2},\ \sqrt{3},\ \sqrt{5},\ \sqrt{7}$ Not surds: $\sqrt{4}=2,\ \sqrt{9}=3,\ \sqrt{16}=4$
⚡ Simplifying Surds
$$\sqrt{ab} = \sqrt{a} \times \sqrt{b}$$Find the largest perfect-square factor, take it out.Example: $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$
🔧 Rationalising the Denominator
Remove surds from the denominator by multiplying numerator and denominator by the surd (or its conjugate).
Simple: $\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$
Conjugate: $\frac{1}{3+\sqrt{2}} \times \frac{3-\sqrt{2}}{3-\sqrt{2}} = \frac{3-\sqrt{2}}{9-2} = \frac{3-\sqrt{2}}{7}$
The conjugate of $(a + \sqrt{b})$ is $(a - \sqrt{b})$. Their product gives the rational number $a^2 - b$.
Ready to test yourself? Click the Quiz tab above to answer questions on this topic!
Interactive Demonstration — Indices & Surds
🎬
Interactive demonstration available in the Calculator tab!
⚡ Indices & Surds Tool