Chapter 1: Number Systems

Rationalizing the Denominator

What is Rationalization?

Rationalization = Making the denominator a rational number

We convert expressions like 12\frac{1}{\sqrt{2}} to equivalent forms without square roots in denominator.

Type 1: Denominator = √a

Method: Multiply by aa\frac{\sqrt{a}}{\sqrt{a}}

Example: Rationalize 12\frac{1}{\sqrt{2}}

Step 1: Multiply by √2/√2 12×22=22\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}

Answer: 22\frac{\sqrt{2}}{2}

Type 2: Denominator = a + √b

Method: Multiply by conjugate a−ba−b\frac{a - \sqrt{b}}{a - \sqrt{b}}

Example: Rationalize 12+3\frac{1}{2 + \sqrt{3}}

Step 1: Identify conjugate: 2 - √3

Step 2: Multiply 12+3×2−32−3\frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}}

Step 3: Simplify denominator using (a+b)(a-b) = a² - b² =2−34−3=2−31=2−3= \frac{2 - \sqrt{3}}{4 - 3} = \frac{2 - \sqrt{3}}{1} = 2 - \sqrt{3}

Type 3: Denominator = √a + √b

Example: Rationalize 53−5\frac{5}{\sqrt{3} - \sqrt{5}}

Step 1: Conjugate = √3 + √5

Step 2: Multiply 53−5×3+53+5\frac{5}{\sqrt{3} - \sqrt{5}} \times \frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}}

Step 3: Simplify =5(3+5)3−5=53+55−2= \frac{5(\sqrt{3} + \sqrt{5})}{3 - 5} = \frac{5\sqrt{3} + 5\sqrt{5}}{-2}

Answer: −53+552-\frac{5\sqrt{3} + 5\sqrt{5}}{2}

Visualizer
Step-by-Step Solution
1
1/√2
Original expression
2
= (1/√2) × (√2/√2)
Multiply by √2/√2 = 1
3
= √2 / (√2 × √2)
Simplify numerator and denominator
4
= √2 / 2
√2 × √2 = 2. Done!

Interactive Visualization