Surd Multiplication Rule

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}