Operations with Radicals

Q: How do I multiply radicals?

Multiply the insides: $\sqrt{a}\cdot\sqrt{b} = \sqrt{ab}$, then simplify the product.

A radical is in simplest form when no perfect-square factor is left inside. Simplifying means pulling those factors out front.

Once simplified, radicals add like terms — only matching radicals combine — and multiply by combining what is inside the roots. Simplifying first makes everything else easier.

Simplifying a radical

Factor the radicand to find its largest perfect-square factor, then take that root out front. $\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}$ .

Keep factoring until nothing inside the root is a perfect square. That is simplest form.

Adding and multiplying

Only like radicals add: $2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$ , but $\sqrt{2} + \sqrt{3}$ cannot be combined. Simplify first so like radicals become obvious.

To multiply, combine the insides: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ , then simplify the result.

Worked examples

Example 1: simplify a radical

Simplify $\sqrt{48}$ .

Find the largest perfect-square factor

48 = 16 \cdot 3

Take the root of 16 out

\sqrt{16}\sqrt{3}

Simplify

4\sqrt{3}

Answer: $4\sqrt{3}$

Example 2: adding like radicals

Add $2\sqrt{5} + 3\sqrt{5}$ .

Same radical, add coefficients

(2 + 3)\sqrt{5}

Simplify

5\sqrt{5}

Answer: $5\sqrt{5}$

Example 3: multiplying radicals

Multiply $\sqrt{6} \cdot \sqrt{2}$ .

Combine what is inside the roots

\sqrt{6 \cdot 2}

Multiply

\sqrt{12}

Take out the perfect-square factor

\sqrt{4}\sqrt{3}

Simplify

2\sqrt{3}

Answer: $2\sqrt{3}$

Try one yourself

Common questions

When is a radical simplified?

When no perfect-square factor remains inside. Pull out the largest perfect-square factor to simplify.

Which radicals can be added?

Only like radicals — same index and same radicand. $2\sqrt{3}$ and $5\sqrt{3}$ combine; $\sqrt{2}$ and $\sqrt{3}$ do not.

How do I multiply radicals?

Multiply the insides: $\sqrt{a}\cdot\sqrt{b} = \sqrt{ab}$ , then simplify the product.

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Operations with Radicals

Simplifying a radical

Adding and multiplying

Worked examples

Example 1: simplify a radical

Example 2: adding like radicals

Example 3: multiplying radicals

Try one yourself

Common questions

Keep learning

nth Roots & Rational Exponents

Graphing Radical Functions

Simplifying Radical Expressions