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

Can you multiply radicals with different radicands?

Yes. Use the product property: √a × √b = √(ab). Multiply the radicands together and then simplify. The result may or may not be a whole number.

What is the conjugate and why is it useful?

The conjugate of \(\color{blue}{(a + \sqrt{b})}\) is \(\color{blue}{(a – \sqrt{b})}\). Multiplying conjugates eliminates the radical using the difference of squares pattern: \(\color{blue}{(a + \sqrt{b})(a – \sqrt{b}) = a^{2} – b}\). This technique is also used when rationalizing denominators.

Must you always simplify after multiplying?

Yes — always check whether the product under the radical can be simplified. A perfect square factor under the radical means the expression is not yet in simplest form.

Related Topics

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