What index notation means
When the same number is multiplied by itself many times, writing it out is slow. Index notation is a short way to record it. In the form aⁿ, the number a is the base and the small raised number n is the index (also called the exponent or power). It tells you how many times the base is used as a factor.
Key idea
aⁿ = a × a × a × ... (n copies of a). For example 3⁴ means 3 × 3 × 3 × 3, so the base is 3 and the index is 4.
Reading and building powers
We read 2⁵ as "2 to the power of 5". To turn a number back into a power, ask what base multiplied by itself gives that number. Since 2 × 2 × 2 × 2 × 2 = 32, we can write 32 = 2⁵.
Worked example
Write 2 × 2 × 2 × 2 in index notation and find its value. Count the copies of 2: there are 4, so it equals 2⁴. Its value is 2 × 2 × 2 × 2 = 4 × 4 = 16. So 2⁴ = 16.
Signs and zero
Watch the base carefully. (−2)³ = (−2) × (−2) × (−2) = −8, because an odd number of negative factors gives a negative answer. A base of 10 is special: 10⁴ = 10 000, which is a 1 followed by 4 zeros.
Remember
- The base is the number being multiplied; the index counts the factors.
- aⁿ is not a × n. 3⁴ = 81, not 12.
- 10ⁿ is 1 followed by n zeros.