When you multiply the same number by itself many times, writing it out becomes long:
\(2 \times 2 \times 2 \times 2 \times 2 = \;?\)
Instead we write \(2^5\) and read it as “two to the fifth power.”
From repeated multiplication to elegant shortcuts. Powers grow very quickly — let’s learn how to tame them.
When you multiply the same number by itself many times, writing it out becomes long:
\(2 \times 2 \times 2 \times 2 \times 2 = \;?\)
Instead we write \(2^5\) and read it as “two to the fifth power.”
\(3^4\) = “three to the fourth power” = \(3 \times 3 \times 3 \times 3 = 81\)
\(5^2\) = “five squared” = \(5 \times 5 = 25\)
\(7^1\) = simply \(7\). Any number to the first power is the number itself.
For olympiad problems these values should be memorised.
| n | \(2^n\) | \(3^n\) | \(5^n\) | \(10^n\) |
|---|---|---|---|---|
| 1 | 2 | 3 | 5 | 10 |
| 2 | 4 | 9 | 25 | 100 |
| 3 | 8 | 27 | 125 | 1 000 |
| 4 | 16 | 81 | 625 | 10 000 |
| 5 | 32 | 243 | 3 125 | 100 000 |
| 6 | 64 | 729 | — | — |
| 7 | 128 | — | — | — |
| 8 | 256 | — | — | — |
| 9 | 512 | — | — | — |
| 10 | 1 024 | — | — | — |
Squares of numbers are useful powers too:
\(1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225\)
These are \(1^2, 2^2, 3^2, \dots, 15^2\).
These rules let us simplify expressions without calculating enormous numbers.
\(2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 2^7\)
3 twos + 4 twos = 7 twos.
CMS problems often ask you to compare powers. Here are three main strategies:
These tasks match the style of real olympiad problems on powers.