Squares and powers
What 13² and 2¹⁰ mean, the squares worth knowing by heart, and a quick way to square numbers like 45 in your head.
A power is a short way to write the same number multiplied by itself many times. 2¹⁰ means ten 2s multiplied together, and the answer, 1024, is why computer memory is often counted in steps of 1024. The power you meet most often is the square: a number multiplied by itself, like 13² = 13 × 13 = 169. This page shows what powers mean, which squares are worth knowing by heart, and quick ways to square numbers in your head.
What a power means
In 3⁵, the big number 3 is called the base. The small raised number 5 is called the exponent, or the power. The exponent tells you how many times the base is written in the multiplication.
Remember
3⁵ = 3 × 3 × 3 × 3 × 3
The exponent counts how many 3s are multiplied, not what 3 is multiplied by.
A number to the power 2 is called squared, and to the power 3 it is called cubed. The name "squared" comes from geometry: a square with a side of 5 has an area of 5 × 5 = 25, so 5² = 25.
Powers of 2, 3 and 5
To work out a power, start with the base and keep multiplying by it, one step at a time. Count the steps carefully.
Example 1
What is 2¹⁰?
- Start with 2 and double it again and again.
- 2, 4, 8, 16, 32 (that is 2⁵).
- 64, 128, 256, 512, 1024 (that is 2¹⁰).
- You wrote ten numbers, so you used ten 2s.
Answer: 1024
Example 2
What is 3⁵?
- Multiply by 3 again and again.
- 3, 9, 27, 81, 243.
- That is five numbers, so five 3s.
Answer: 243
| Exponent | 1 | 2 | 3 | 4 | 5 | 10 |
|---|---|---|---|---|---|---|
| Base 2 | 2 | 4 | 8 | 16 | 32 | 1024 |
| Base 3 | 3 | 9 | 27 | 81 | 243 | 59,049 |
| Base 5 | 5 | 25 | 125 | 625 | 3125 | 9,765,625 |
Multiplying powers with the same base
What is 2⁵ × 2³? Write it out: 2⁵ is five 2s and 2³ is three 2s. Multiplied together, that is eight 2s, so the answer is 2⁸ = 256.
Remember
a^m × aⁿ = a^(m+n)
Only when the base is the same. Add the exponents.
Squares worth knowing
Squares appear everywhere: in areas, in right triangles, and in many quick tricks. These are worth learning by heart.
| Number | Square | Number | Square |
|---|---|---|---|
| 11 | 121 | 16 | 256 |
| 12 | 144 | 17 | 289 |
| 13 | 169 | 18 | 324 |
| 14 | 196 | 19 | 361 |
| 15 | 225 | 20 | 400 |
If you forget one, you can always work it out like any other multiplication: split one of the numbers into tens and ones.
Example 3
What is 14²?
- 14² = 14 × 14.
- Split the second 14 into 10 + 4.
- 14 × 10 = 140 and 14 × 4 = 56.
- 140 + 56 = 196.
Answer: 196
Squaring numbers that end in 5
There is a lovely trick for numbers like 25, 35 and 45. Take the tens digit and multiply it by the next number up. Then write 25 after the answer.
- Take the tens digit (for 45, it is 4).
- Multiply it by the next number up: 4 × 5 = 20.
- Write 25 after it: 2025.
Example 4
What is 35²?
- The tens digit is 3. The next number up is 4.
- 3 × 4 = 12.
- Write 25 after it: 1225.
- Check: 35 × 35 = 30 × 40 + 25 = 1200 + 25 = 1225.
Answer: 1225
The check shows why the trick works: 35 × 35 is the same as 30 × 40 plus 25. In the same way, 25² = 20 × 30 + 25 = 625.
Numbers near a round number
When a number is close to a round number, multiply by the round number and then fix the difference.
Example 5
What is 99²?
- 99 × 99 is 99 hundreds take away one 99.
- 99 × 100 = 9900.
- 9900 − 99 = 9801.
Answer: 9801
There is also a pattern when two numbers sit the same distance either side of a round number, like 21 and 19 (one each side of 20). Their product is the square of the middle number, take away the square of the distance.
Example 6
What is 52 × 48?
- 52 and 48 are both 2 away from 50.
- The middle number squared: 50² = 2500.
- The distance squared: 2² = 4.
- 2500 − 4 = 2496.
Answer: 2496
Try it yourself
1.What is 18²?
Show the answer
324
18 × 10 = 180 and 18 × 8 = 144, so 180 + 144 = 324.
2.What is 3⁴?
Show the answer
81
Multiply by 3 four times: 3, 9, 27, 81. It is not 3 × 4 = 12.
3.What is 65²?
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4225
The tens digit is 6, and 6 × 7 = 42. Write 25 after it: 4225.
4.What is 41 × 39?
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1599
Both are 1 away from 40. 40² = 1600, take away 1² = 1: 1599.
5.What is 2⁴ × 2³?
Show the answer
128
The base is the same, so add the exponents: 2⁷ = 128. Check: 16 × 8 = 128.
6.What is 51²?
Show the answer
2601
2500 + 2 × 50 × 1 + 1 = 2500 + 100 + 1 = 2601.
Ready to test yourself?
Play Arithmetic: 20 seconds per question, and it gets harder as you go.