• To avoid confusion, when writing large numbers or small decimal fractions, group the digits in threes from the decimal point. E.g. 49 560 216 and 0.003 48.
• A number in standard form is written as a × 10ⁿ, where a is a number between 1 and 11 and n is a positive or negative integer.
• The laws of indices are:
  • y^a × y^b = y^a+b
  • x^a ÷ x^b = x^a-b
  • x⁰ = 1
  • x^-a = ¹/x^a
  • (x^a)^b = x^ab

Examples

1. Express the following numbers in standard form:
   1. 60 000
      • = 6 × 10 000
      • = 6 × 10⁴
   2. 480 000 000
      • = 4.8 × 100 000 000
      • = 4.8 × 10⁸
2. Express the following in ordinary form:
   1. 4.3 × 10⁴
      • = 4.3 × 10 000
      • = 43 000
   2. 7.8 × 10⁷
      • = 7.8 × 10 000 000
      • = 78 000 000
3. Write the following as decimal fractions:
   1. 350 millionths
      • = 1 millionth × 350
      • = 0.000 001 × 350
      • = 0.000 350
      • = 0.000 35
      • NOTE: In a decimal fraction, it is not necessary to write any zeros after the last non-zero digit.
   2. ⁸⁶⁵/_{100 000}
      • = 0.008 65
      • NOTE: There are five zeros in the denominator. The decimal fraction is obtained by moving the digits in the numerator five places to the right.
4. Simplify the following:
   1. 10⁴ × 10²
      • = (10 × 10 × 10 × 10) × (10 × 10)
      • = 10 × 10 × 10 × 10 × 10 × 10
      • = 10⁶
      • OR more quickly, by adding indices: 10⁴ × 10² = 10⁴⁺² = 10⁶
   2. 4c³ × 7c²
      • = 4 × c × c × c × 7 × c × c
      • = 4 × 7 × c × c × c × c × c
      • = 28c⁵
      • OR more quickly, by adding indices: 4c³ × 7c² = 4 × 7 × c³⁺² = 28c⁵
5. Simplify the following:
   1. 10⁴ ÷ 10²
      • Using one of the laws of indices:
      • = 10^{4 – 2}
      • ∴ 10⁴ ÷ 10² = 10²
   2. 4c³ ÷ 7c²
      • ⁴/₇ × c^3-2
      • ∴ 4c³ ÷ 7c² = ⁴/₇C