Example 1: Dividing Two Radicals

Problem: Divide $8\sqrt{15}$ by $2\sqrt{3}$.

Solution:

To solve this, we can rewrite the number inside the square root ($\sqrt{15}$) into its factors to see if anything cancels out.

1. Set up the fraction:
   $$\frac{8\sqrt{15}}{2\sqrt{3}}$$
2. Factor the radical: Since $15 = 5 \times 3$, we can write $\sqrt{15}$ as $\sqrt{5} \times \sqrt{3}$.
   $$\frac{8 \times \sqrt{5} \times \sqrt{3}}{2\sqrt{3}}$$

3. Simplify:

   • Divide the whole numbers: $8 \div 2 = 4$.

   • Cancel out the common radical: $\sqrt{3} \div \sqrt{3} = 1$.

4. Final Result:
   $$4\sqrt{5}$$

Example 2: Dividing a Whole Number by a Radical

Problem: Divide $8$ by $2\sqrt{2}$.

Solution:

This example uses a specific property of radicals: any positive number $a$ can be written as $\sqrt{a} \times \sqrt{a}$.

1. Set up the fraction and simplify the coefficients:
   $$\frac{8}{2\sqrt{2}} = \frac{4}{\sqrt{2}}$$
2. Rewrite the numerator: To divide by $\sqrt{2}$, we can express the number $2$ as $\sqrt{2} \times \sqrt{2}$.
   $$\frac{2 \times 2}{\sqrt{2}}$$
3. Apply the radical rule:
   $$\frac{2 \times \sqrt{2} \times \sqrt{2}}{\sqrt{2}}$$
4. Cancel and Solve: One $\sqrt{2}$ in the numerator cancels with the $\sqrt{2}$ in the denominator.

   Final Result:

   $$2\sqrt{2}$$

Key Rule to Remember:

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