VCE Mathematical Methods (Year 11) · Unit 1 — Algebra & Structure · 11m read

Index Laws and Exponential Expressions

Ever wondered why 2³ × 2⁴ = 2⁷ works out so neatly? Index laws are the shortcut rules that let you simplify expressions with powers and roots without expanding everything. You'll use these constantly in VCE maths—from solving exponential equations to working with surds (those tricky square roots)—and they're essential for calculus and physics too.

Learning objective: Manipulate indices and surds fluently.

By the end of this lesson you can:

• I apply index laws to simplify expressions.
• I rationalise denominators.
• I convert between exponential and surd form.

Index laws are exponent shortcuts. They let you combine powers without writing out the long multiplication — like using a calculator for repeated steps.

Index laws

THE CORE RULES:
• a^m × a^n = a^(m+n)
• a^m ÷ a^n = a^(m−n)
• (a^m)^n = a^(mn)
• (ab)^n = a^n b^n
• (a/b)^n = a^n / b^n
• a^0 = 1 (a ≠ 0)
• a^(−n) = 1 / a^n
• a^(1/n) = ⁿ√a
• a^(m/n) = ⁿ√(a^m)

SURDS:
• A SURD is an irrational number with a root, e.g., √2, √5.
• √a × √b = √(ab).
• √a ÷ √b = √(a/b).
• √a + √a = 2√a (same surd terms combine).

SIMPLIFYING SURDS:
• Factor out the largest perfect square.
• E.g., √72 = √(36 × 2) = 6√2.

RATIONALISING DENOMINATORS:
• a/√b → multiply top and bottom by √b.
• a/(b + √c) → multiply by conjugate (b − √c).

EXPONENTIAL ↔ SURD:
• √x = x^(1/2).
• ³√(x²) = x^(2/3).
• 1/x³ = x^(−3).

COMMON ERRORS:
• (a + b)^n ≠ a^n + b^n.
• (a^m)^n ≠ a^(m+n). It's a^(mn).
• √(a + b) ≠ √a + √b.

APPLICATION: EXPONENTIAL FUNCTIONS:
• y = a × b^x grows (b > 1) or decays (0 < b < 1).
• Used for compound interest, radioactive decay, population.

COMPOUND INTEREST:
• A = P(1 + r/n)^(nt).
• Continuous compounding → A = Pe^(rt) (later in the course).

Index quick wins

• Negative power: Flip the fraction.
• Fractional: Root of a power.
• Zero power: Always 1.

Index laws

• Multiply: a^m a^n = a^{m+n}
• Divide: a^m / a^n = a^{m-n}
• Negative index: a^{-n} = 1/a^n
• Fractional index: a^{1/n} = \sqrt[n]{a}

Quick reference

| Law | Rule |
|---|---|
| Product | aᵐaⁿ = aᵐ⁺ⁿ |
| Quotient | aᵐ/aⁿ = aᵐ⁻ⁿ |
| Power | (aᵐ)ⁿ = aᵐⁿ |
| Negative | a⁻ⁿ = 1/aⁿ |
| Fractional | a^(m/n) = ⁿ√(aᵐ) |
| Zero | a⁰ = 1 |

Worked example (Medium)

Simplify: (2a²b³)³ × (a⁻¹b²)² ÷ (ab)⁴.

1. Expand each term. (2a²b³)³ = 8a⁶b⁹. (a⁻¹b²)² = a⁻²b⁴. (ab)⁴ = a⁴b⁴.
2. Multiply numerator. 8a⁶b⁹ × a⁻²b⁴ = 8a⁴b¹³.
3. Divide by denominator. 8a⁴b¹³ ÷ a⁴b⁴ = 8b⁹.

Answer: 8b⁹.

Worked example (Hard)

Rationalise and simplify: $\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}$.

1. Multiply by conjugate. $\frac{(\sqrt{3} + \sqrt{2})(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})}$.
2. Denominator. $(\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1$.
3. Numerator. $(\sqrt{3} + \sqrt{2})^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6}$.

Answer: $5 + 2\sqrt{6}$.

Common mistakes

• Claiming $(a + b)^n = a^n + b^n$ Students incorrectly distribute the exponent over addition Fix: Exponent laws only apply to multiplication/division of bases: $(ab)^n = a^n b^n$ but $(a+b)^n$ must be expanded.
• Writing $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ Confusion between the product rule $\sqrt{ab} = \sqrt{a}\sqrt{b}$ and addition Fix: Test with numbers: $\sqrt{9+16} = 5 \neq 3 + 4 = 7$. The rule only works for products.
• Confusing $(a^m)^n = a^{m+n}$ with the correct $a^{mn}$ Mixing up the power-of-a-power rule with the multiplication rule Fix: Power of a power MULTIPLIES indices. Multiplying same-base terms ADDS indices.

Surds, irrationals, and why rationalising matters

A surd is an irrational root that cannot be simplified to a rational number — $\sqrt{2}$, $\sqrt{5}$, $\sqrt[3]{7}$. While your calculator happily displays decimals, VCAA demands exact answers, making surd manipulation a core skill.

Rationalising the denominator isn't just mathematical tradition — it serves practical purposes. When the denominator is rational, you can more easily compare, add, and simplify expressions. The conjugate technique exploits the difference-of-squares identity: $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$, eliminating the surd from the denominator.

Fractional indices bridge surds and exponentials. Writing $\sqrt[3]{x^2}$ as $x^{2/3}$ unlocks index laws for simplification and is essential for calculus, where you differentiate $x^{2/3}$ using the power rule but cannot differentiate $\sqrt[3]{x^2}$ directly.

In VCE Methods, surd fluency underpins exact values of trigonometric ratios ($\sin 60° = \frac{\sqrt{3}}{2}$), solutions to quadratics via the formula ($x = \frac{-b \pm \sqrt{\Delta}}{2a}$), and later, integration results. Students who avoid surds and reach for the calculator lose marks on tech-free papers.

A powerful practice habit: after every simplification, check with a calculator that the decimal value of your simplified form matches the original.

Practice questions

1. Simplify 4^(3/2).

   1. 6
   2. 8
   3. 12
   4. 16

   (1 marks)

2. Rationalise: 3 / (2 + √5).

   (3 marks)

Summary

• Apply index laws to simplify expressions.
• Rationalise denominators.
• Convert between exponential and surd form.