Power Play

Overview

This chapter introduces exponents and powers as a short way to write repeated multiplication. We study the laws of exponents, negative and zero exponents, and how very large or very small numbers are written in standard (scientific) form.

Key concepts

• a^m means a multiplied by itself m times; a is the base, m is the exponent.
• Any non-zero number raised to the power 0 equals 1.
• A negative exponent means a reciprocal: a^−m = 1 ÷ a^m.
• Standard form writes a number as k × 10ⁿ, where 1 ≤ k < 10.

Important formulae

• a^m × aⁿ = a^m+n
• a^m ÷ aⁿ = a^m−n
• (a^m)ⁿ = a^mn
• (a × b)^m = a^m × b^m

Solved example

1. Simplify 2³ × 2⁻⁵.
2. Add exponents: 2^{3 + (−5)} = 2⁻².
3. Convert: 2⁻² = 1 ÷ 2² = 1 ÷ 4.

Important questions

1. Express 0.0000256 in standard form.
2. Simplify (3²)³ ÷ 3⁴.
3. Find the value of (−2)⁻³.
4. The mass of an electron is about 9 × 10⁻³¹ kg. Write it in ordinary decimal form.

Quick revision

Use the laws of exponents to combine powers with the same base. A zero exponent gives 1; a negative exponent gives a reciprocal. Standard form is a clean way to handle very big or very small numbers.