Powers
Multiplying many times without writing so much
What is this about?
A power is a short way of writing a repeated multiplication. In 2⁵, the 2 is the base (the number that repeats) and the 5 is the exponent (how many times it's multiplied). So 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. It's not a new operation: it's ordinary multiplication written in compact mode.
The rule to remember
The base is multiplied by itself as many times as the exponent says. Special cases you must know: any number raised to 1 is itself, and any number (other than 0) raised to 0 is 1.
Example 1
3^4
- 1The base is 3 and the exponent is 4: you multiply the 3 four times.
- 2Go step by step: 3 × 3 = 9.
- 3Keep going: 9 × 3 = 27.
- 4One more time: 27 × 3 = 81.
Result: 81
Example 2
2^3 × 2^2
- 1Both powers have the same base (2), so they can be combined.
- 2When multiplying powers with the same base, the exponents ARE ADDED: 3 + 2 = 5.
- 3That leaves 2⁵ = 2 × 2 × 2 × 2 × 2.
- 4Calculate: 2⁵ = 32.
Result: 32
The most common mistake
Many people write
2^3 = 6
Why it's wrong
The base was multiplied by the exponent (2 × 3) instead of multiplying the base by itself. The exponent doesn't multiply: it counts how many times the base repeats. The correct way is 2 × 2 × 2 = 8.
Did it click?
The best way to find out is to try. These exercises are exactly on this topic.
Practise this topic