Surd Division Rule

The surd division rule lets you simplify the division of two square roots (or other roots with the same index) by combining them under a single radical. In its basic form, the rule is:

How It Works

1. Same Index Requirement:
   The rule applies when both the numerator and denominator are radicals with the same index (commonly square roots).

2. \[ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \]
3. \[ \frac{k\sqrt{m}}{l\sqrt{n}} \]\[ \frac{k}{l} \times \frac{\sqrt{m}}{\sqrt{n}} = \frac{k}{l} \times \sqrt{\frac{m}{n}} \]

Example

So, \(\frac{3\sqrt{50}}{5\sqrt{2}} = 3\).

Important Points

• Domain Restrictions:
  Make sure \(b\) (or \(n\) in the general form) is not zero, and the values under the radicals are non-negative when dealing with real numbers.

• Rationalizing the Denominator:
  Sometimes you’ll want to remove the radical from the denominator (a process called rationalizing), but using the surd division rule often simplifies this process.

This rule is a powerful tool for simplifying expressions that involve radicals and is commonly used in algebra to make further calculations easier.

Latex

\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \quad \text{(provided } b \neq 0\text{)}