Powers: base, exponent and powers of 10 — Mathematics, 11–13
A power as a product of equal factors: base and exponent, powers of 10, and why repeated multiplication grows so fast.
A power: multiplying the same number several times
Instead of writing 3 × 3 × 3 × 3, you write 3⁴ (read ‘three to the power of four’). The 3 is the base — the factor that repeats — and the 4 is the exponent — how many times it appears. 3⁴ = 3 × 3 × 3 × 3 = 81. A power is a product of equal factors, and both the base and the exponent are natural numbers.
Why powers grow so fast
Notice the difference between adding and multiplying the same number several times. With ten 2s: 2 + 2 + … + 2 = 20, but 2 × 2 × … × 2 = 2¹⁰ = 1024. Each new factor multiplies the result instead of adding a little to it: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. That is why, with a natural base greater than 1, a power grows faster and faster as the exponent goes up.
Powers of base 10: 10, 100, 1000, 10 000
10 = 10¹. 100 = 10 × 10 = 10². 1000 = 10 × 10 × 10 = 10³. 10 000 = 10 × 10 × 10 × 10 = 10⁴. Notice: the exponent is the number of zeros after the 1. It also works backwards: 10⁵ is a 1 followed by five zeros, 100 000; and 10⁶ is 1 000 000. So a huge number becomes short: 1 000 000 = 10⁶, much quicker than counting zeros.
The trap: multiplying the base by the exponent
The most common mistake: reading 2⁵ as 2 × 5 = 10. But 2⁵ = 2 × 2 × 2 × 2 × 2 = 32: the exponent tells you how many times the 2 is written, not what to multiply by. In the same way, 10³ is not 30: it is 10 × 10 × 10 = 1000. A hint to avoid the slip: a power with base 2 or more and exponent 3 or more is always bigger than base × exponent. If your answer is small, be suspicious.
Where you see powers outside school
In a social-media challenge, each person invites 3 new people who have not been invited yet. In round 1 that is 3 people, in round 2 it is 3² = 9, in round 3 it is 3³ = 27, in round 4 it is 3⁴ = 81, and in round 6 already 3⁶ = 729. News that spreads this way reaches many people very quickly. If everyone invited 2 instead of 3, round 6 would give only 2⁶ = 64.
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