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Index laws and standard form
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Index laws are shortcuts for working with powers. Standard form uses powers of 10 to write very big and very small numbers neatly.
Index laws
Multiplying: add the powers. a5 × a3 = a8
Dividing: subtract the powers. a9 ÷ a4 = a5
Power of a power: multiply. (x2)3 = x6
Anything to the power 0 is 1: 70 = 1.
A negative power means a reciprocal: 2−3 = 123 = 18.
Dividing: subtract the powers. a9 ÷ a4 = a5
Power of a power: multiply. (x2)3 = x6
Anything to the power 0 is 1: 70 = 1.
A negative power means a reciprocal: 2−3 = 123 = 18.
Standard form
A number in standard form looks like A × 10n, where 1 ≤ A < 10 and n is an integer.
45 000 = 4.5 × 104 (the digits move 4 places)
0.0032 = 3.2 × 10−3 (small numbers have a negative power)
45 000 = 4.5 × 104 (the digits move 4 places)
0.0032 = 3.2 × 10−3 (small numbers have a negative power)
Calculating
Multiply the numbers, then use the index laws on the powers of 10.
(3 × 104) × (2 × 103) = 6 × 107
If A ends up 10 or more, adjust: 12 × 105 = 1.2 × 106.
(3 × 104) × (2 × 103) = 6 × 107
If A ends up 10 or more, adjust: 12 × 105 = 1.2 × 106.
Write 0.000 52 in standard form.
- Move the digits until you get 5.2
- That is 4 places, and the number is small, so the power is negative
Answer: 5.2 × 10−4
Add powers when multiplying, subtract when dividing, multiply for a power of a power. Standard form is A × 10n with A from 1 up to (but not including) 10.
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