Radicals

A root left exact

$\sqrt{2}$√2 is the number that makes 2 when it is multiplied by itself. It is not a whole number, because 1 × 1 = 1 is too small and 2 × 2 = 4 is too big. Decimals get closer: 1.4 × 1.4 = 1.96 and 1.42 × 1.42 = 2.0164, so $\sqrt{2}$√2 is between 1.4 and 1.42. And 1.41 × 1.41 = 1.9881, so it is between 1.41 and 1.42.

You can keep going forever. The decimal for $\sqrt{2}$√2 never stops and never repeats, so every decimal you write, such as 1.414, is only an approximation. Writing $\sqrt{2}$√2 itself is the exact value. A root written this way, because its decimal never ends, is called a radical (in many countries, a surd).

The decimals close in on $\sqrt{2}$√2 from both sides, but none of them ever lands on it.

The root of a product

Radicals can often be written in a simpler way, and the reason is a rule about products: the square root of a product is the product of the square roots. For example, $\sqrt{4\times 9}=\sqrt{36}=6$√(4 × 9) = √36 = 6, and $\sqrt{4}\times \sqrt{9}=2\times 3=6$√4 × √9 = 2 × 3 = 6 as well.

The rule holds for every pair of numbers. Multiply $\sqrt{4}\times \sqrt{3}$√4 × √3 by itself: $\sqrt{4}\times \sqrt{3}\times \sqrt{4}\times \sqrt{3}=\left(\sqrt{4}\times \sqrt{4}\right)\times \left(\sqrt{3}\times \sqrt{3}\right)=4\times 3=12$√4 × √3 × √4 × √3 = (√4 × √4) × (√3 × √3) = 4 × 3 = 12. So $\sqrt{4}\times \sqrt{3}$√4 × √3 is the number whose square is 12, which means $\sqrt{12}=\sqrt{4}\times \sqrt{3}$√12 = √4 × √3.

Take the square factor out

To simplify $\sqrt{12}$√12, split 12 into a square number times whatever is left. 12 = 4 × 3, and 4 is a square number, so $\sqrt{12}=\sqrt{4}\times \sqrt{3}$√12 = √4 × √3.

$\sqrt{4}$√4 is exactly 2, so the square part comes out of the root as a whole number, and the 3 stays inside: $\sqrt{12}=2\sqrt{3}$√12 = 2√3. The $2\sqrt{3}$2√3 means $2\times \sqrt{3}$2 × √3, just as 2x means 2 × x.

A square of area 12 cut into four equal squares. Each small square has area 3, so its side is $\sqrt{3}$√3, and the big square's side is two of those: $2\sqrt{3}$2√3.

Use the largest square factor

Look for the largest square number that divides the number. For $\sqrt{50}$√50, the largest square factor is 25: 50 = 25 × 2, so $\sqrt{50}=\sqrt{25}\times \sqrt{2}=5\sqrt{2}$√50 = √25 × √2 = 5√2.

For $\sqrt{72}$√72, you might split 72 as 4 × 18, which gives $\sqrt{72}=2\sqrt{18}$√72 = 2√18. But 18 still holds the square factor 9, so the work goes on: $\sqrt{18}=\sqrt{9}\times \sqrt{2}=3\sqrt{2}$√18 = √9 × √2 = 3√2, and $2\times 3\sqrt{2}=6\sqrt{2}$2 × 3√2 = 6√2. Using the largest square factor, 36, gets there in one step: $\sqrt{72}=\sqrt{36}\times \sqrt{2}=6\sqrt{2}$√72 = √36 × √2 = 6√2. A radical is in its simplest form when the number left inside has no square factor other than 1.

Check by squaring

Square the answer to check it. ${\left(2\sqrt{3}\right)}^2=2\times 2\times \sqrt{3}\times \sqrt{3}=4\times 3=12$(2√3)² = 2 × 2 × √3 × √3 = 4 × 3 = 12, the number you started with, so $\sqrt{12}=2\sqrt{3}$√12 = 2√3 is right. In the same way, ${\left(5\sqrt{2}\right)}^2=25\times 2=50$(5√2)² = 25 × 2 = 50, and ${\left(6\sqrt{2}\right)}^2=36\times 2=72$(6√2)² = 36 × 2 = 72.

Common mistakes

The square factor comes out as its root, not as itself. $\sqrt{12}$√12 is not $4\sqrt{3}$4√3: squaring $4\sqrt{3}$4√3 gives 16 × 3 = 48, not 12. The 4 must come out as $\sqrt{4}=2$√4 = 2.

Nothing spare stays inside the root. $\sqrt{12}$√12 is not $2\sqrt{6}$2√6: squaring $2\sqrt{6}$2√6 gives 4 × 6 = 24. Once the 4 has come out as 2, only the 3 is left inside.

A sum never splits this way. $\sqrt{9}+\sqrt{16}=3+4=7$√9 + √16 = 3 + 4 = 7, but $\sqrt{9+16}=\sqrt{25}=5$√(9 + 16) = √25 = 5. The rule is for products only.