### Square roots of non-perfect squares

We can write the number 40 as 2 × 2 × 2 × 5. This is not a perfect square because we cannot pair the factors.

We write \(\sqrt{40}\) to be the square root of 40, and can investigate between which two whole numbers it lies.

Consider the table of perfect squares below:

Whole number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Whole number squared | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |

The number 40 lies between 36 and 49 hence \(\sqrt{40}\) lies between 6 and 7.

Similarly, 12 lies between 9 and 16 so \(\sqrt{12}\) lies between 3 and 4.

These numbers can be shown approximately on the number line.