Operations with Radicals

Q: Can I add the numbers under the roots, like $\sqrt{4} + \sqrt{9} = \sqrt{13}$?

No. $\sqrt{4} + \sqrt{9} = 2 + 3 = 5$, but $\sqrt{13}$ is about $3.6$. Roots don't distribute over addition — combining insides only works for multiplication.

A radical is a root, like a square root. This lesson covers the three moves you need: simplifying a radical (pulling a perfect square out from under the root), adding or subtracting radicals (only like radicals combine), and multiplying radicals (the insides combine under one root).

The whole topic runs on one short list: the perfect squares $4, 9, 16, 25, 36, 49, 64, 81, 100$ . Know those cold and every problem here becomes a factoring search.

Simplify: pull out the perfect square

Simplifying a radical means finding the largest perfect-square factor hiding under the root and pulling its root out front. Since $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ , you can split $\sqrt{12}$ into $\sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$ .

Always hunt for the largest perfect-square factor. $\sqrt{72}$ with the factor $4$ gives $2\sqrt{18}$ — not finished, because $18$ still hides a $9$ . Starting with $36$ gets you to $6\sqrt{2}$ in one step.

Add and subtract: only like radicals combine

Radicals add the way variable terms do. Just as $3x + 5x = 8x$ , $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$ — the number under the root has to match, and then you add the coefficients. $3\sqrt{2} + 5\sqrt{3}$ can't be combined, the same way $3x + 5y$ can't.

Sometimes radicals that look different are secretly like radicals. $\sqrt{8} + 3\sqrt{2}$ doesn't combine as written, but simplify first: $\sqrt{8} = 2\sqrt{2}$ , and now $2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$ . Simplify every radical before deciding nothing combines.

Multiply: combine under one root

To multiply radicals, merge the insides: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ . So $\sqrt{6} \cdot \sqrt{15} = \sqrt{90}$ .

Then simplify the result: $90 = 9 \cdot 10$ , so $\sqrt{90} = 3\sqrt{10}$ . Multiplying radicals almost always ends with one more simplifying step — don't stop at the merged root.

Worked examples

Example 1: simplify a radical

Simplify $\sqrt{72}$ .

Find the largest perfect-square factor of

72

72 = 36 \cdot 2

Split the root

\sqrt{72} = \sqrt{36} \cdot \sqrt{2}

Take the root of the perfect square

6\sqrt{2}

Answer: $6\sqrt{2}$

Example 2: add like radicals

Simplify $2\sqrt{3} + 4\sqrt{3}$ .

Same number under the root, so these are like radicals

Add the coefficients

2 + 4 = 6

Keep the radical

6\sqrt{3}

Answer: $6\sqrt{3}$

Example 3: multiply, then simplify

Simplify $\sqrt{6} \cdot \sqrt{15}$ .

Combine under one root

\sqrt{6 \cdot 15} = \sqrt{90}

Find the perfect-square factor

90 = 9 \cdot 10

Pull it out

3\sqrt{10}

Answer: $3\sqrt{10}$

Try one yourself

Common questions

How do I know when a radical is fully simplified?

When the number under the root has no perfect-square factor left other than $1$ . $6\sqrt{2}$ is done because $2$ hides no perfect square; $2\sqrt{18}$ is not, because $18 = 9 \cdot 2$ .

Why can't I add

3\sqrt{2} + 5\sqrt{3}

For the same reason $3x + 5y$ won't combine — the radical parts are different objects. Only radicals with the same number under the root act like the same variable.

Can I add the numbers under the roots, like

\sqrt{4} + \sqrt{9} = \sqrt{13}

No. $\sqrt{4} + \sqrt{9} = 2 + 3 = 5$ , but $\sqrt{13}$ is about $3.6$ . Roots don't distribute over addition — combining insides only works for multiplication.

Do the perfect squares really matter that much?

Yes — they're the whole toolkit. Every simplify step is a search for a perfect-square factor, so knowing $4, 9, 16, 25, 36, 49, 64, 81, 100$ by heart makes these problems fast.

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