How to Multiply Radical Expressions? (+FREE Worksheet!)

Multiplying radical expressions uses the same product property that makes simplifying radicals possible: √a × √b = √(ab). Once you know this rule, you can multiply single radical terms, multiply a radical by a binomial, and even multiply two binomials containing radicals using FOIL. This guide covers every case with four worked examples and two video lessons.

What Is the Product Property of Radicals?

For any non-negative values a and b:

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√a × √b = √(\(\color{blue}{a \times b}\))

Multiply what is under the radical signs together, then simplify the result if possible. Coefficients outside the radical are multiplied separately, just like ordinary numbers.

Rules for Multiplying Radical Expressions

1. Multiply Coefficients Together

Treat the whole-number parts in front of the radicals as ordinary factors: \(\color{blue}{3\sqrt{5} \times 2\sqrt{5} = (3 \times 2)\sqrt{(5 \times 5)} = 6 \times 5 = 30}\).

2. Multiply the Radicands Together

\(\color{blue}{\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6}\)

3. Simplify the Result

After multiplying radicands, simplify the resulting radical if possible. \(\color{blue}{\sqrt{(6 \times 10)} = \sqrt{60} = \sqrt{(4 \times 15)} = 2\sqrt{15}}\).

4. Multiplying Binomials with Radicals

Use the FOIL method or distributive property: \(\color{blue}{(2 + \sqrt{3})(1 – \sqrt{3}) = 2 – 2\sqrt{3} + \sqrt{3} – 3 = -1 – \sqrt{3}}\).

Step-by-Step Summary

1. Multiply the coefficients (the numbers outside the radicals).
2. Multiply the radicands (the numbers or expressions inside the radicals).
3. Simplify the resulting radical by finding perfect square factors.
4. Combine any like radical terms if the expression has been expanded.

Watch: Multiplying Radical Expressions (Video Lesson)

Khan Academy walks through multiplying and then simplifying the resulting radical expression:

Multiplying Radical Expressions — Worked Examples

Example 1: Multiply \(\color{blue}{\sqrt{3} \times \sqrt{12}}\).

Multiply radicands: \(\color{blue}{\sqrt{(3 \times 12)} = \sqrt{36} = 6}\).
Answer: \(\color{blue}{6}\)

Example 2: Multiply \(\color{blue}{3\sqrt{5} \times 2\sqrt{5}}\).

Coefficients: \(\color{blue}{3 \times 2 = 6}\). Radicands: \(\color{blue}{\sqrt{(5 \times 5)} = \sqrt{25} = 5}\).
Answer: \(\color{blue}{6 \times 5 = 30}\)

Example 3: Multiply \(\color{blue}{\sqrt{6} \times \sqrt{10}}\).

Radicands: \(\color{blue}{\sqrt{(6 \times 10)} = \sqrt{60} = \sqrt{(4 \times 15)} = 2\sqrt{15}}\).
Answer: \(\color{blue}{2\sqrt{15}}\)

Example 4: Multiply \(\color{blue}{(2 + \sqrt{3})(1 – \sqrt{3})}\).

Use FOIL:
F: \(\color{blue}{2 \times 1 = 2}\)
O: \(\color{blue}{2 \times (-\sqrt{3}) = -2\sqrt{3}}\)
I: \(\color{blue}{\sqrt{3} \times 1 = \sqrt{3}}\)
L: \(\color{blue}{\sqrt{3} \times (-\sqrt{3}) = -3}\)
Combine: \(\color{blue}{2 – 2\sqrt{3} + \sqrt{3} – 3 = -1 – \sqrt{3}}\)

More Practice: Step-by-Step Video

This video covers multiplying radical expressions with additional examples including binomials:

Exercises for Multiplying Radical Expressions

Find each product in simplest form.

1. \(\color{blue}{\sqrt{2} \times \sqrt{18}}\)
2. \(\color{blue}{2\sqrt{3} \times 5\sqrt{3}}\)
3. \(\color{blue}{\sqrt{5} \times \sqrt{20}}\)
4. \(\color{blue}{(\sqrt{2} + 1)(\sqrt{2} – 1)}\)
5. \(\color{blue}{3\sqrt{2} \times 4\sqrt{6}}\)

Answers

1. \(\color{blue}{6}\)  (\(\color{blue}{\sqrt{36} = 6}\))
2. \(\color{blue}{30}\)  (coefficients \(\color{blue}{2 \times 5 = 10}\); \(\color{blue}{\sqrt{9} = 3}\); \(\color{blue}{10 \times 3 = 30}\))
3. \(\color{blue}{10}\)  (\(\color{blue}{\sqrt{100} = 10}\))
4. \(\color{blue}{1}\)  (difference of squares: \(\color{blue}{(\sqrt{2})^{2} – 1^{2} = 2 – 1 = 1}\))
5. \(\color{blue}{24\sqrt{3}}\)  (coefficients: \(\color{blue}{3 \times 4 = 12}\); radicand: \(\color{blue}{\sqrt{12} = 2\sqrt{3}}\); \(\color{blue}{12 \times 2\sqrt{3} = 24\sqrt{3}}\))

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Want More Practice?

We haven’t published a worksheet built specifically for Multiplying Radical Expressions just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work. They’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Multiplying Polynomials Worksheet
• Download Special Products of Polynomials Worksheet
• Download Graphing Square Root Functions Worksheet

Frequently Asked Questions

How do you multiply radical expressions?

Multiply radicals with the same index by multiplying the radicands, using √a × √b = √(ab), and then simplify. For √6 × √15, multiply to get √90. Since 90 = 9 × 10, √90 simplifies to 3√10. If there are numbers in front, multiply those separately: 2√3 × 5√2 = 10√6.

How do you multiply radical expressions with variables?

Multiply the coefficients, multiply what is under the radicals, and then take out perfect squares, including even powers of variables. For √(2x) × √(8x³), the product is √(16x⁴). Because 16 = 4² and x⁴ = (x²)², this simplifies to 4x². For square roots, assume the variables are nonnegative so each step is defined.

How do you multiply radical expressions with different index?

Rewrite each radical as a rational exponent, find a common denominator for the exponents, and add them. For √2 × ∛2, write 2^(1/2) × 2^(1/3). Since 1/2 + 1/3 = 5/6, the product is 2^(5/6), which is the sixth root of 2⁵, or ⁶√32. This works only when the bases match or can be rewritten to match.

When multiplying radical expressions, you must have the same what?

You must have the same index, meaning both are square roots, both are cube roots, and so on. The radicands do not need to match: √3 × √5 = √15 works even though 3 and 5 differ. That is the opposite of addition, where like radicals need the same radicand too. If the indexes differ, change to rational exponents before multiplying.

What is an example of multiplying radical expressions?

A good example with two binomials is (2 + √3)(4 − √3), which you expand with FOIL. First: 2 × 4 = 8. Outer: 2 × (−√3) = −2√3. Inner: √3 × 4 = 4√3. Last: √3 × (−√3) = −3. Combine 8 − 3 = 5 and −2√3 + 4√3 = 2√3 to get 5 + 2√3.

Related Topics

• Simplifying Radical Expressions
• Adding and Subtracting Radical Expressions
• Rationalizing Radical Expressions
• Radical Equations
• Domain and Range of Radical Functions