How to simplify radicals (simplest radical form)

A radical is a root sign with a number (or expression) under it, like $\sqrt{50}$ or $\sqrt[3]{54}$. The number underneath is the radicand, and the small number in the crook of the sign is the index — no index means a square root. Learning how to simplify radicals means rewriting them so the number under the sign is as small as possible.

The key fact is that, for numbers that aren’t negative, the root of a product is the product of the roots: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. If part of the radicand is a perfect square, its root comes out as a whole number.

What is simplest radical form?

A radical expression is in simplest radical form (also called simplified radical form) when:

1. the radicand has no perfect-square factor other than 1 (for a cube root, no perfect-cube factor);
2. there is no fraction under the radical sign;
3. there is no radical left in the denominator of a fraction.

How to simplify radicals, step by step

To simplify a square root, pull out the biggest perfect square hiding inside it:

1. Find the largest perfect square that divides the radicand.
2. Write the radicand as that square times what is left.
3. Take the square root of the perfect square and put it in front of the radical.
4. Check that what is left under the sign has no square factors.

Perfect squares worth knowing

Spotting the largest perfect square is much faster if you know the squares by heart: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225$ (the squares of 1 to 15). For cube roots, the first perfect cubes are $1, 8, 27, 64, 125, 216$.

They also help you estimate. $\sqrt{50}$ lies between $\sqrt{49} = 7$ and $\sqrt{64} = 8$, and much closer to 7 — so $5\sqrt{2} \approx 7.07$ makes sense.

Radicals with a number in front

When a radical already has a coefficient, simplify the radical and multiply what comes out by that number. Radical form is the exact answer; a decimal is only an approximation.

How to simplify radical expressions

Radicals behave like algebra terms: you can add or subtract only like radicals — the same index and the same radicand — just as $4y + 3y = 7y$. So simplify each radical first, then collect like ones. For example, $2\sqrt{12} + \sqrt{27} = 2(2\sqrt{3}) + 3\sqrt{3} = 4\sqrt{3} + 3\sqrt{3} = 7\sqrt{3}$. (Check: $2\sqrt{12} + \sqrt{27} \approx 6.928 + 5.196 = 12.124$, and $7\sqrt{3} \approx 12.124$.)

Subtracting works the same way: $5\sqrt{8} - \sqrt{18} = 5(2\sqrt{2}) - 3\sqrt{2} = 10\sqrt{2} - 3\sqrt{2} = 7\sqrt{2}$. (Check: $14.142 - 4.243 = 9.899$, and $7\sqrt{2} \approx 9.899$.)

Multiplying is easier: multiply the numbers outside and the numbers inside, then simplify. $\sqrt{6} \times \sqrt{15} = \sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}$.

With a radical in the denominator, multiply the top and bottom by that radical (this is called rationalising the denominator): $\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$. A fraction under the sign splits the same way: $\sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{\sqrt{4}} = \frac{\sqrt{3}}{2}$.

Variables work the same way, assuming they stand for numbers that aren’t negative: $\sqrt{50x^3} = \sqrt{25x^2 \times 2x} = 5x\sqrt{2x}$.

Common mistakes

• Splitting a sum: $\sqrt{9 + 16} = \sqrt{25} = 5$, not $\sqrt{9} + \sqrt{16} = 7$. The root of a product splits; the root of a sum does not.
• Stopping too early: $\sqrt{72} = 2\sqrt{18}$ is true but not finished, because 18 still has the square factor 9. Use the largest square, 36, to get $6\sqrt{2}$.
• Adding unlike radicals: $\sqrt{2} + \sqrt{3}$ is not $\sqrt{5}$ — it can’t be simplified.
• Leaving a radical in the denominator when the question asks for simplest radical form.