IP AMaths Notes (Upper Sec, Year 3-4): 01) Indices and Surds

Master the index laws so exponent and surd manipulations become automatic. Keep every expression in exact form until the last step to avoid rounding drift.

Essential identities

• \( a^{m} \times a^{n} = a^{m+n} \), \( \dfrac{a^{m}}{a^{n}} = a^{m-n} \) for \( a \neq 0 \).
• \( \bigl(a^{m}\bigr)^{n} = a^{mn} \); extend to rationals so \( a^{\dfrac{p}{q}} = \sqrt[q]{a^{p}} \).
• \( \sqrt{ab} = \sqrt{a}\sqrt{b} \) for \( a, b \geq 0 \); rationalise denominators with conjugates.
• Standard rationalisations: \( \dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a} \) and \( \dfrac{1}{a + \sqrt{b}} = \dfrac{a - \sqrt{b}}{a^{2} - b} \).

Worked example 1 — Simplify and rationalise

Simplify \( \dfrac{5x^{3}\sqrt{18x}}{3x\sqrt{2x}} \) for \( x > 0 \).

1. Combine the surds: \( \sqrt{18x} = \sqrt{9 \times 2x} = 3\sqrt{2x} \).
2. Substitute: \( \dfrac{5x^{3} \times 3\sqrt{2x}}{3x\sqrt{2x}} = \dfrac{15x^{3}\sqrt{2x}}{3x\sqrt{2x}} \).
3. Cancel like factors: \( \dfrac{15x^{3}\sqrt{2x}}{3x\sqrt{2x}} = 5x^{2} \).

Answer: \( 5x^{2} \).

Worked example 2 — Solve power equation

Solve \( 27^{x-1} = 9^{2x} \).

1. Express both sides with base 3: \( 27 = 3^{3} \), \( 9 = 3^{2} \).
2. Rewrite: \( \bigl(3^{3}\bigr)^{x-1} = \bigl(3^{2}\bigr)^{2x} \) so \( 3^{3(x-1)} = 3^{4x} \).
3. Equate indices: \( 3(x - 1) = 4x \) giving \( 3x - 3 = 4x \).
4. Solve: \( x = -3 \).

Verify: \( 27^{-4} = 9^{-6} \) — true because both equal \( 3^{-12} \).

Try this

Show that if \( a, b > 0 \) and \( \sqrt{a} + \sqrt{b} = 5 \) with \( ab = 6 \), then \( a + b = 19 \).