Index notation

Objective. Write repeated multiplication using powers and evaluate integer powers.

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Learn: the key idea

Index notation writes repeated multiplication of the same number compactly using a base and a power, such as 3⁴ meaning 3 × 3 × 3 × 3. The small raised number is the index (or exponent) and tells you how many times the base is multiplied by itself. Any non-zero number raised to the power 0 equals 1, and a power of 1 leaves the number unchanged, so 6⁰ = 1 and 6¹ = 6. To evaluate 2⁵, multiply 2 by itself five times to get 32. Index notation becomes essential later for standard form and algebraic powers.

Worked example

Write 2 × 2 × 2 × 2 × 2 in index notation and evaluate it.

1 $Count the number of twos being multiplied: five$
2 $Write in index notation: 2⁵$
3 $Multiply out: 2 \times 2 \times 2 \times 2 \times 2$
4 $Evaluate: 2⁵ = 32$

Answer: 32

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Common mistake. Pupils often calculate 3⁴ as 3 × 4 = 12 instead of multiplying 3 by itself four times; the correct value is 3 × 3 × 3 × 3 = 81. Always write out the repeated multiplication to avoid this.

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