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Indices


3³ ('3 cubed' or '3 to the power of 3') and 5² ('5 squared' or 5 'to the power' of 2) are example of numbers in index form.
3³ = 3×3×3
2¹ = 2
2² = 2×2
2³ = 2×2×2
etc.
The ² and ³ are known as indices. Indices are useful (for example they allow us to represent numbers in standard form) and have a number of properties.
Laws of Indices

There are several rules for dividing and multiplying numbers written in index form. These properties only hold, however, when the same number is being raised to a certain power. For example, we cannot easily work out what 2³×5² is, whereas we can simplify 3²×3³ .
Multiplication

When we multiply together index numbers, we add the powers. So:
ya × yb = ya+b
Examples

x2 × x3 = x5
54 × 5-2 = 52 (because 4 + (-2) = 2)
But there is no easy way of calculating 54 × 33 because 5 and 3 aren't the same number!
Division

When dividing index numbers, we subtract the power of the number we are dividing by from the power of the number being divided. So:
ya ÷ yb = ya - b
Examples

x2/x3 = x-1
72 ÷ 7-5 = 77
Brackets

(ya)b = ya×b
Examples

(x2)3 = x6
(53)2 = 56



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