Evaluate 16−3/4.
- The negative sign means the reciprocal: 16−3/4 = 1 ÷ 163/4.
- The 4 means the fourth root: the fourth root of 16 is 2, because 24 = 16.
- The 3 means cube it: 23 = 8, so the value is 1 ÷ 8.
Answer: 1/8
Indices I
Maths · Number. 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 (power) is a short way to write repeated multiplication: a4 = a × a × a × a. The rules of indices let you combine powers of the same base without working out the large numbers.
Indices do not have to be positive whole numbers. Any non-zero number to the power 0 is 1. A negative index means “one over”: it gives the reciprocal, not a negative number. A fractional index means a root: the denominator tells you which root and the numerator tells you which power.
With a fractional index, take the root first. The numbers stay small and the answer is easier to find.
Rules to remember
- am × an = am + n
- am ÷ an = am − n
- (am)n = am × n
- a0 = 1 (for a not equal to 0)
- a−n = 1 ÷ an, and (a/b)−n = (b/a)n
- a1/n = the nth root of a, so 91/2 = 3 and 81/3 = 2
- am/n = (nth root of a)m
Common mistake
Thinking a negative index gives a negative answer. 2−3 is 1/8, not −8 and not −6.
Worked example
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