This lesson builds on your Grade 10 knowledge by introducing fractional indices and a new logarithm law. Mastering these will make complex-looking problems easy to solve.

1. Core Concept: Fractional Indices (Rational Indices)

Fractional indices are just a different way of writing roots.

a) Basic Fractional Indices: The Root

The number on the bottom of the fraction is the root.

• Formula: an1​=na​

• Example: 2731​=327​=3

• Example: 1621​=16​=4

b) Advanced Fractional Indices: Root and Power

• Formula: anm​=(na​)m

• How to solve:

  1. Deal with the root first (the denominator n).

  2. Then apply the power (the numerator m).

• Example: Solve

  1. Apply the root (cube root): 36427​​=43​

  2. Apply the power (square): (43​)2=169​

c) Negative Fractional Indices: Flip the Fraction!

A negative index means "find the reciprocal".

• Formula: a−n=an1​

• How to solve:

  1. Flip the base fraction to make the index positive.

  2. Solve the fractional index as normal (root, then power).

• Example: Solve (8116​)−43​

  1. Flip the base: (1681​)43​

  2. Apply the root (fourth root): 41681​​=23​

  3. Apply the power (cube): (23​)3=827​

2. Solving Equations with Indices

This is a guaranteed exam question type. The goal is always the same.

Method: Make the bases the same, then equate the powers. If am=an, then m=n.

• Example: Solve 3×92x−1=27−x

  1. Find a common base: The common base for 3, 9, and 27 is 3.

  2. Rewrite all terms with the common base:

     • 3=31

     • 9=32

     • 27=33

  3. Substitute back into the equation: 31×(32)2x−1=(33)−x

  4. Use index laws to simplify each side to a single power:

     • Law 1: (am)n=amn -> 31×32(2x−1)=33(−x) -> 31×34x−2=3−3x

     • Law 2: am×an=am+n -> 31+(4x−2)=3−3x -> 34x−1=3−3x

  5. Equate the powers and solve: 4x−1=−3x 7x=1 x=71​

3. The "Power Rule" of Logarithms

You know that log(A) + log(B) = log(AB) and log(A) - log(B) = log(A/B). The new rule lets you handle powers.

• Formula: loga​(mr)=rloga​m

• In simple terms: You can bring the power down to the front as a multiplier. This also works for roots!

• Example (Power): log5​(1254)=4log5​(125)=4×3=12

• Example (Root): log4​(364​)=log4​(6431​)=31​log4​(64)=31​×3=1

4. Solving Equations with Logarithms

Method: Use the log laws to get a single log on each side, then drop the logs. If loga​M=loga​N, then M=N.

• Example: Solve 2logb​3+3logb​2−logb​72=21​logb​x

  1. Use the Power Rule first to move all multipliers up into the powers: logb​(32)+logb​(23)−logb​72=logb​(x21​)

  2. Use addition/subtraction rules to combine the left side into one log term: logb​(7232×23​)=logb​(x21​) logb​(729×8​)=logb​(x21​) logb​(7272​)=logb​(x21​) logb​(1)=logb​(x21​)

  3. Drop the logs and solve the remaining equation: 1=x21​ 1=x​ x=1

Exam Tips & Common Mistakes

• Mistake 1 (Indices): Rushing the calculation. Always do the root first, then the power. For 832​, find 38​ (which is 2) before you square it.

• Mistake 2 (Logs): Confusing the log rules. log A + log B is log(AB), NOT log(A+B). You cannot simplify log(A+B).

• Exam Tip (Indices): When solving equations, immediately look for the common prime base (usually 2, 3, or 5). Break down every number in the equation into that base.

• Exam Tip (Logs): When solving log equations, your first step should always be to apply the power rule to get rid of any numbers in front of the log terms.