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Year 7AC9M7N02

Index Notation and Powers

Represent natural numbers as products of powers of prime numbers using exponent notation.

Index notation is shorthand for repeated multiplication. It saves writing, and it makes the patterns in numbers visible.

Method

The base and the index
25=2×2×2×2×22^5 = 2\times2\times2\times2\times2
The base is what you multiply. The index counts how many times.
An index is not a multiplier
25≠2×52^5 \neq 2 \times 5
This is the most common error. $2^5 = 32$, not 10.
Squares and cubes have names
32=933=273^2 = 9 \quad 3^3 = 27
Squared means index 2. Cubed means index 3.

Worked example

e.g. Write 3×3×3×3 in index form and find its value.\text{Write } 3\times3\times3\times3 \text{ in index form and find its value.}
  1. Count how many times the number appears.3 appears 4 times3 \text{ appears } 4 \text{ times}
  2. The number is the base, the count is the index.343^4
  3. To find the value, multiply it out step by step.3×3=93\times3 = 9
  4. Keep going one factor at a time.9×3=2727×3=819\times3 = 27 \quad 27\times3 = 81
  5. Write the answer.34=813^4 = 81

Practice

Write in index form, then find the value.

1
2×2×22\times2\times2
Answer23=82^3 = 8
2
5×55\times5
Answer52=255^2 = 25
3
10410^4
Answer10 00010\,000
4
Which is larger: 24 or 42?\text{Which is larger: } 2^4 \text{ or } 4^2?
AnswerEqual — both are 16\text{Equal — both are } 16
5
Why is 32≠6?\text{Why is } 3^2 \neq 6?
Answer32 means 3×3=9, not 3×23^2 \text{ means } 3\times3 = 9, \text{ not } 3\times2
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