How to Multiply and Divide Surds

In this lesson, we teach you how to multiply and divide surds using simple rules. If you can simplify surds, then multiplying and dividing them is a straightforward process.

If you don't know what a surd is, then you might want to read our introduction to surds first.

How to Multiply Surds

In general, surd multiplication follows this rule:

\(\sqrt{a} \times \sqrt{b} = \sqrt{a \cdot b}\)

Eg. \(\sqrt{7} \times \sqrt{3} = \sqrt{7 \cdot 3} = \sqrt{21}\)

If the surds have numbers in front of them (coefficients), multiply the coefficients separately:

\(a\sqrt{b} \times c\sqrt{d} = ac\sqrt{bd}\)

Eg. \(2\sqrt{5} \times 8\sqrt{7} = 16\sqrt{35}\)

Sometimes the resulting surd can be simplified.

Eg. \(3\sqrt{6} \times \sqrt{8} = 3\sqrt{48}\ = 3\sqrt{16}\sqrt{3} = 12\sqrt{3} \)

Simple Examples

Example 1

Question: \(\sqrt{11} \times \sqrt{5}\)

Answer: \(\sqrt{55}\)

Example 2

Question: \(\sqrt{6} \times 3\sqrt{7}\)

Answer: \(3\sqrt{42}\)

Example 3

Question: \(5\sqrt{2} \times 2\sqrt{10}\)

Answer: \(10\sqrt{20} = 10\sqrt{4}\sqrt{5} = 20\sqrt{5} \)

Simplifying Before Multiplying

If you can simplify the surds, do so before multiplying. This can make calculations easier.

Example 1: \(\sqrt{24} \times 3\sqrt{5}\)

Simplify \(\sqrt{24}\) to \(2\sqrt{6}\):

\(2\sqrt{6} \times 3\sqrt{5} = 6\sqrt{30}\)

Example 2: \(\sqrt{60} \times 7\sqrt{3}\)

Simplify \(\sqrt{60}\) to \(2\sqrt{15}\):

\(2\sqrt{15} \times 7\sqrt{3} = 14\sqrt{45}\)

\( \sqrt{45}\) can also be simplified to \( 3\sqrt{5} \)

So \( 14\sqrt{45} = 14 \times 3\sqrt{5} = 42\sqrt{5} \)

How to Divide Surds

In general, surd division follows this rule:

\(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)

Eg. \(\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3\)

Examples

Example 1

Question: \(\sqrt{54} \div \sqrt{6}\)

\(\sqrt{54} \div \sqrt{6} = \sqrt{\frac{54}{6}} = \sqrt{9} = 3\)

Answer: \(3\)

Example 2

Question: \(\frac{6\sqrt{50}}{2\sqrt{2}}\)

\(\frac{6\sqrt{50}}{2\sqrt{2}} = \frac{6}{2} \sqrt{\frac{50}{2}} = 3\sqrt{25} = 3 \times 5 = 15\)

Answer: \(15\)

If you can simplify the surd first, this can make calculations easier.

Example 3

Question: \(\frac{\sqrt{32}}{\sqrt{18}}\)

First, simplify each surd:

\( \sqrt{32} = 4\sqrt{2} \)

\( \sqrt{18} = 3\sqrt{2} \)

So \(\frac{\sqrt{32}}{\sqrt{18}}\) becomes \(\frac{4\sqrt{2}}{3\sqrt{2}} = \frac{4}{3}\)

Answer: \( \frac{4}{3}\)