Square Root Calculator and Chart

$\sqrt{n}$ asks: what non-negative number, multiplied by itself, gives $n$? When $n$ is a perfect square the answer is a whole number, and knowing these ten by heart is what makes every other square root question fast:

Beyond 100 the pattern continues: $11^2 = 121$, $12^2 = 144$, $13^2 = 169$, $14^2 = 196$, $15^2 = 225$.

Everything else is irrational — $\sqrt{2} = 1.41421\ldots$ never terminates or repeats. The chart is still the tool you use, because the exact form of $\sqrt{72}$ is found by hunting for the largest perfect square hiding inside 72.

Reading the chart backwards. To place $\sqrt{60}$, find the two perfect squares it sits between: $49 < 60 < 64$, so the answer lies between 7 and 8, and closer to 8. That bracketing trick is enough to sanity-check any calculator result before you trust it.

Two rules do almost all the work:

$\sqrt{ab} = \sqrt{a}\,\sqrt{b} \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

To simplify $\sqrt{n}$:

1. Find the largest perfect square that divides $n$. For 72 that is 36, since $72 = 36 \times 2$.
2. Split the root: $\sqrt{72} = \sqrt{36}\sqrt{2}$.
3. Take the square root of the perfect square and leave the rest inside: $6\sqrt{2}$.

If you cannot spot the largest square, prime factorise instead and pull out each pair: $72 = 2^3 \times 3^2$, so one pair of 2s and one pair of 3s come out as $2 \times 3 = 6$, leaving a single 2 behind.

Like radicals add. $4\sqrt{3} + 3\sqrt{3} = 7\sqrt{3}$, exactly as $4x + 3x = 7x$. Unlike radicals, such as $\sqrt{2} + \sqrt{3}$, cannot be combined at all.

• The root does not distribute over addition. $\sqrt{9 + 16} = \sqrt{25} = 5$, but $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$. It splits over multiplication and division only.
• $\sqrt{25}$ is 5, not $\pm 5$. The radical symbol means the principal (non-negative) root. The $\pm$ appears when you solve $x^2 = 25$ yourself, because that equation has two roots.
• Stopping at a square you can still pull out. $\sqrt{72} = 2\sqrt{18}$ is true but not simplified — 18 still contains a factor of 9.
• Adding unlike radicals. $\sqrt{2} + \sqrt{3} \neq \sqrt{5}$; check numerically: $1.414 + 1.732 = 3.146$, while $\sqrt{5} \approx 2.236$.
• Squaring away a root without checking. Squaring both sides of $\sqrt{x} = -3$ gives $x = 9$, but $\sqrt{9} = 3 \neq -3$, so it is extraneous.