🔢 Power Up Your Algebra: Conquer Indices II!

Indices (also called powers) let us write repeated multiplication compactly. **Mastering index laws** is vital for simplifying algebraic expressions quickly, a skill that O-Level exam questions love to test.

Key Index Laws Refresher

Remember these **golden rules** for any base

a \neq 0

Product rule: $a^{m}\,a^{n}=a^{m+n}$
Quotient rule: $\dfrac{a^{m}}{a^{n}}=a^{m-n}$
Power of a power: $(a^{m})^{n}=a^{mn}$
Negative index: $a^{-m}=\dfrac{1}{a^{m}}$
Zero index: $a^{0}=1$

Worked Example: Simplifying Algebraic Indices

Problem: Simplify

(3a^{2}b^{-1})^{3} \div 9a^{-1}b^{5}

Step 1: Expand the power $(3a^{2}b^{-1})^{3} = 3^{3} a^{2\times3} b^{-1\times3} = 27a^{6}b^{-3}$
Step 2: Write the full expression as a fraction: $\dfrac{27a^{6}b^{-3}}{9a^{-1}b^{5}}$
Step 3: Simplify constants: $27\div9=3$
Step 4: Apply quotient rule to $a$ terms: $a^{6}/a^{-1}=a^{6-(-1)}=a^{7}$
Step 5: Apply quotient rule to $b$ terms: $b^{-3}/b^{5}=b^{-3-5}=b^{-8}$
Final answer: $3a^{7}b^{-8}=\dfrac{3a^{7}}{b^{8}}$

Practice makes perfect! **Challenge yourself** by creating similar problems and timing how fast you can simplify them. The more fluent you are with index laws, the more exam marks you will secure.