Key Concepts:
Simplifying Surds:
Break the number inside the root into factors, one of which is a perfect square: \[ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \]Multiplying and Dividing Surds:
\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{ab}, \quad \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \]Rationalizing the Denominator:
Multiply numerator and denominator by a suitable surd to eliminate a root in the denominator: \[ \frac{1}{\sqrt{3}} = \frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{3} \]Using Conjugates:
To rationalize denominators with two terms: \[ \frac{1}{a + \sqrt{b}} = \frac{1 \cdot (a - \sqrt{b})}{(a + \sqrt{b})(a - \sqrt{b})} \]Examples:
Example 1: Simplify \(\sqrt{72}\)
Solution:
Step 1: Break into perfect square and other factor: \[ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \]Example 2: Rationalize \(\frac{5}{\sqrt{2}}\)
Solution:
Step 1: Multiply numerator and denominator by \(\sqrt{2}\): \[ \frac{5}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{2} \]Example 3: Rationalize \(\frac{3}{2+\sqrt{3}}\)
Solution:
Step 1: Multiply numerator and denominator by the conjugate \(2 - \sqrt{3}\): \[ \frac{3}{2+\sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{3(2 - \sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})} \] Step 2: Simplify: \[ \text{Numerator: } 6 - 3\sqrt{3} \] Denominator: 4 - 3 = 1 \[ \] Final answer: \[ 6 - 3\sqrt{3} \]