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:

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

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. Combining Radicals:
   Since (\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}), division works similarly by taking the quotient inside the radical:

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

3. General Form:
   If you have coefficients (numbers in front of the radicals), say:

   $$\frac{k\sqrt{m}}{l\sqrt{n}}$$

   you can write this as:

   $$\frac{k}{l} \times \frac{\sqrt{m}}{\sqrt{n}} = \frac{k}{l} \times \sqrt{\frac{m}{n}}$$

Example

Simplify:

$$\frac{3\sqrt{50}}{5\sqrt{2}}$$

Step 1: Separate the coefficients and the radicals:

$$\frac{3}{5} \times \frac{\sqrt{50}}{\sqrt{2}}$$

Step 2: Apply the surd division rule to the radicals:

$$\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25}$$

Step 3: Simplify the radical:

$$\sqrt{25} = 5$$

Step 4: Multiply back the coefficient:

$$\frac{3}{5} \times 5 = 3$$

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{)}