What is Rationalization?
Rationalization is the process of eliminating irrational numbers (like √2, √3) from the denominator of a fraction. This makes expressions easier to work with and compare.
Why Rationalize?
- To simplify expressions
- To add/subtract fractions with irrational denominators
- To compare numbers easily
- Standard form in mathematics often requires rationalized denominators
Methods for Rationalization:
| Type of Denominator | Multiply Numerator & Denominator by | Result |
|---|---|---|
| a + √b | a - √b (conjugate) | Denominator becomes a² - b |
| √a + √b | √a - √b (conjugate) | Denominator becomes a - b |
The Conjugate Trick:
The conjugate of (a + √b) is (a - √b). Their product:
(a + √b)(a - √b) = a² - (√b)² = a² - b (a rational number)