O Level MathematicsC1.7 Indices I (positive, zero, and negative integer indices).

🔢 Power Up! Mastering Positive, Zero & Negative Indices 💡

Edudent Academy
29 Nov 25

Indices (or powers) let us write repeated multiplication compactly—vital when dealing with very large or tiny numbers in O-Level questions. **A firm grip on index laws unlocks faster algebraic manipulation and boosts exam speed!**

Core Rules of Indices

  • am×an=am+na^m \times a^n = a^{m+n}
    (Product rule)
  • aman=am−n  (a≠0)\dfrac{a^m}{a^n}=a^{m-n}\;(a\neq0)
    (Quotient rule)
  • (am)n=amn(a^m)^n=a^{mn}
    (Power of a power)
  • a0=1  (a≠0)a^0 = 1\;(a\neq0)
    (Zero index)
  • a−n=1ana^{-n}=\dfrac{1}{a^{n}}
    (Negative index)

Worked Example: Simplify 2−3×252^{-3}\times2^{5}

Problem: Simplify 2−3×252^{-3}\times2^{5}.\n\nStep 1: Apply the product rule:
2−3×25=2(−3+5)=222^{-3}\times2^{5}=2^{(-3+5)}=2^{2}
\nStep 2: Evaluate the remaining positive power:
22=42^{2}=4
\n\nTherefore, 2−3×25=42^{-3}\times2^{5}=4. **Notice how the negative exponent became positive once combined—always combine powers before converting negatives to fractions to save time!**\n\nKeep practising different combinations of positive, zero, and negative indices so these steps feel automatic on exam day.