Rationalize Denominator Calculator

Remove radicals from the denominator of a fraction

√

Denominator form: a√b

Other Titles
Understanding Denominator Rationalization: A Comprehensive Guide
Learn the process of rewriting a fraction to eliminate radical expressions from its denominator, making it simpler and easier to work with.

What Does it Mean to Rationalize the Denominator?

Rationalizing the denominator is a conventional process in mathematics to remove a radical (like a square root) from the denominator of a fraction. While a fraction with a radical in the denominator is mathematically correct, it's considered 'unsimplified' in many contexts. The goal is to convert the fraction into an equivalent form where the denominator is a rational number (an integer).
This is done by multiplying both the numerator and the denominator of the fraction by a carefully chosen expression that will eliminate the radical in the denominator. Because you multiply both the top and bottom by the same value, the overall value of the fraction remains unchanged.

Basic Example

  • Consider the fraction 1/√2.
  • To rationalize it, we multiply the top and bottom by √2.
  • (1/√2) * (√2/√2) = (1 * √2) / (√2 * √2)
  • Since √2 * √2 = 2, the result is √2 / 2.
  • The denominator is now a rational number (2).

Step-by-Step Guide to Using the Calculator

This calculator can handle two common types of denominators with radicals.
1. For Monomial Denominators (like a√b):
2. For Binomial Denominators (like a ± b√c):

Usage Tips

  • Make sure your inputs are valid numbers.
  • The calculator automatically simplifies the final fraction by finding the greatest common divisor.

Methods of Rationalization

Case 1: Monomial Denominator (a√b)
If the denominator is a single term with a square root, you multiply the numerator and denominator by that square root. This removes the radical because √b * √b = b.
Fraction: N / (a√b) => Multiply by √b/√b
Result: (N √b) / (a b)
Case 2: Binomial Denominator (a + √b or a - √b)
If the denominator is a sum or difference involving a square root, you multiply the numerator and denominator by its conjugate.
When you multiply a binomial by its conjugate, you use the difference of squares formula: (x + y)(x - y) = x² - y². This is the key to eliminating the radical.
(a + √b)(a - √b) = a² - (√b)² = a² - b

Conjugate Example

  • Rationalize 2 / (3 + √5).
  • 1. The conjugate of the denominator is (3 - √5).
  • 2. Multiply top and bottom by the conjugate:
  • [2 / (3 + √5)] * [(3 - √5) / (3 - √5)]
  • 3. Numerator: 2 * (3 - √5) = 6 - 2√5
  • 4. Denominator: (3 + √5)(3 - √5) = 3² - (√5)² = 9 - 5 = 4
  • 5. Result: (6 - 2√5) / 4
  • 6. Simplify the fraction by dividing all terms by 2: (3 - √5) / 2

Common Misconceptions

Misconception 1: Only multiply the denominator.
A common mistake is to only multiply the denominator by the radical or conjugate. To keep the fraction's value the same, you must multiply both the numerator and the denominator by the same non-zero value.
Misconception 2: Incorrectly applying the conjugate.
Remember that (a+b)² is not a² + b². Similarly, when dealing with conjugates, you must distribute correctly. The difference of squares formula is a reliable shortcut.

Key Takeaways

  • If the denominator is √b, multiply by √b/√b.
  • If the denominator is a ± √b, multiply by its conjugate (a ∓ √b) / (a ∓ √b).
  • Always simplify the final fraction if possible.

Mathematical Derivation and Examples

The entire process is based on the identity property of multiplication, which states that multiplying any number by 1 does not change its value. The expressions we use, like √b/√b or (a-√b)/(a-√b), are simply clever forms of 1.

Comprehensive Binomial Example

  • Rationalize the fraction: (x + 1) / (4 - 2√3)
  • 1. **Identify the Denominator:** 4 - 2√3
  • 2. **Find the Conjugate:** The conjugate is 4 + 2√3.
  • 3. **Multiply Numerator and Denominator by the Conjugate:**
  • [(x + 1) * (4 + 2√3)] / [(4 - 2√3) * (4 + 2√3)]
  • 4. **Calculate the New Numerator (using FOIL):**
  • (x*4) + (x*2√3) + (1*4) + (1*2√3) = 4x + 2x√3 + 4 + 2√3
  • 5. **Calculate the New Denominator (using difference of squares):**
  • 4² - (2√3)² = 16 - (2² * (√3)²) = 16 - (4 * 3) = 16 - 12 = 4.
  • 6. **Combine and write the final fraction:**
  • (4x + 2x√3 + 4 + 2√3) / 4
  • 7. **Simplify by dividing by common factors if possible.** In this case, we can factor out a 2 from some terms, but to divide the whole expression, all terms must be divisible by the same number. We can rewrite it as: x + (x/2)√3 + 1 + (1/2)√3