The surd multiplication rule states that when you multiply two square roots (surd form), you can combine them into one square root. In mathematical terms:

$$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$

Key Points

• Non-negative Numbers: This rule applies when (a) and (b) are non-negative.
• Rationalizing: After combining, if the product under the square root can be simplified (like being a perfect square), you should simplify it further.

Example

$$\sqrt{2} \times \sqrt{8} = \sqrt{2 \times 8} = \sqrt{16} = 4$$

This property works because of the exponent rule: since (\sqrt{a} = a^{1/2}) and (\sqrt{b} = b^{1/2}), their product is

$$a^{1/2} \times b^{1/2} = (a \times b)^{1/2} = \sqrt{a \times b}.$$

This is the surd multiplication rule!

Latex

\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}