Standard form and indices are among the most searched IGCSE maths topics because they appear across algebra, number, and science-style questions. You need to use laws of indices ((a^m \times a^n = a^{m+n}), (a^m \div a^n = a^{m-n}), ((a^m)^n = a^{mn}), (a^0 = 1), (a^{-n} = 1/a^n), (a^{1/n} = \sqrt[n]{a})), and standard form ((A \times 10^n), (1 \leq A < 10)).

Laws of Indices – Quick Reference

• Multiply same base: add powers. Divide: subtract powers. Power of a power: multiply powers.
• (a^0 = 1); (a^{-n} = 1/a^n); (a^{1/2} = \sqrt{a}); (a^{1/n} = \sqrt[n]{a}).
• Simplify before substituting (e.g. write in single power of 2 or 3).

Standard Form

• Writing: One digit before decimal, rest after; (\times 10^n) (n positive for large numbers, negative for small).
• Calculating: Use calculator (EXP or EE key) or convert and use laws of indices; give final answer in standard form unless asked otherwise.
• Significant figures: Often “give to 3 s.f.” or “in standard form to 2 s.f.”

Practise mixed questions (indices + standard form) so you are confident in exams.

Worked Examples and Common Mistakes

Indices: Simplify 2³ × 2⁵ = 2⁸ (add powers). (3²)⁴ = 3⁸ (multiply powers). 5⁰ = 1. 2⁻³ = 1/2³ = 1/8. Standard form: 34000 = 3.4 × 10⁴. 0.0021 = 2.1 × 10⁻³. When calculating, use your calculator’s EXP or EE key or work in standard form and use laws of indices. Common mistakes: Wrong power (e.g. adding instead of multiplying when raising a power to a power); standard form with more than one digit before the decimal (e.g. 34 × 10³ should be 3.4 × 10⁴); wrong number of significant figures in the final answer. Give the answer in the form requested (e.g. “in standard form to 2 s.f.”).

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