By definition, if $a^2 = b$, then $a$ is called the square root of $b$.
The square root of $b$ is denoted by the symbol $\sqrt{b}$.
The following list demonstrates the square root of various numbers by showing the two identical factors that create the square:
$1^2 = 1 \implies \sqrt{1} = 1$
$2^2 = 4 \implies \sqrt{4} = 2$
$3^2 = 9 \implies \sqrt{9} = 3$
$4^2 = 16 \implies \sqrt{16} = 4$
$5^2 = 25 \implies \sqrt{25} = \sqrt{5 \times 5} = 5$
$\sqrt{36} = \sqrt{6 \times 6} = 6$
$\sqrt{49} = \sqrt{7 \times 7} = 7$
$\sqrt{64} = \sqrt{8 \times 8} = 8$
$\sqrt{81} = \sqrt{9 \times 9} = 9$
$\sqrt{100} = \sqrt{10 \times 10} = 10$
$\sqrt{121} = \sqrt{11 \times 11} = 11$
$\sqrt{144} = \sqrt{12 \times 12} = 12$
$\sqrt{169} = \sqrt{13 \times 13} = 13$
$\sqrt{196} = \sqrt{14 \times 14} = 14$
$\sqrt{225} = \sqrt{15 \times 15} = 15$
$\sqrt{256} = \sqrt{16 \times 16} = 16$
$\sqrt{289} = \sqrt{17 \times 17} = 17$
| Number (b) | Square Root (b) | Number (b) | Square Root (b) |
| 324 | 18 | 576 | 24 |
| 361 | 19 | 625 | 25 |
| 400 | 20 | 676 | 26 |
| 441 | 21 | 729 | 27 |
| 484 | 22 | 784 | 28 |
| 529 | 23 | 841 | 29 |
| 900 | 30 |
There is a distinct pattern when calculating the square roots of larger numbers that end in 25. Notice how the square root always ends in 5:
$\sqrt{225} = 15$
$\sqrt{625} = 25$
$\sqrt{1225} = 35$
$\sqrt{2025} = 45$
$\sqrt{3025} = 55$
$\sqrt{4225} = 65$
$\sqrt{5625} = 75$
$\sqrt{7225} = 85$
$\sqrt{9025} = 95$
When working with large multiples of ten, the number of zeros in the square root is exactly half the number of zeros in the original number:
100 (2 zeros) $\implies \sqrt{10 \times 10} = \mathbf{10}$ (1 zero)
10,000 (4 zeros) $\implies \sqrt{100 \times 100} = \mathbf{100}$ (2 zeros)
1,000,000 (6 zeros) $\implies \sqrt{1000 \times 1000} = \mathbf{1000}$ (3 zeros)