Simplify (2x3)4 ÷ (8x−2).
- (2x3)4 = 24 × x3 × 4 = 16x12
- Numbers: 16 ÷ 8 = 2
- Powers of x: x12 ÷ x−2 = x12 − (−2) = x14
Answer: 2x14
Indices II
Maths · Algebra and graphs. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
An index (or power) tells you how many times a number or letter is multiplied by itself, so x4 = x × x × x × x. The rules of indices let you combine powers of the same base quickly.
Indices do not have to be positive whole numbers. A zero index always gives 1. A negative index means "one over": it gives a reciprocal, not a negative number. A fractional index means a root: the bottom of the fraction is the root and the top is the power.
When a term has a number and a letter, deal with them separately. The numbers are multiplied, divided or raised to the power in the usual way, and the index rules are used for the letters.
Rules to remember
- am × an = am + n
- am ÷ an = am − n
- (am)n = amn and (ab)n = anbn
- a0 = 1 (a not zero)
- a−n = 1/an
- a1/n = the nth root of a, so a1/2 = √a
- am/n = (nth root of a)m, for example 82/3 = 22 = 4
Common mistake
In a term such as (3a2)3, students forget to raise the number to the power, or they multiply it by the power. The 3 must be cubed as well: (3a2)3 = 27a6.
Worked example
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