Rationalizing Denominators Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Rationalize 43+5\frac{4}{3 + \sqrt{5}}.

Solution

  1. 1
    Step 1: Multiply by the conjugate: 43+5⋅3−53−5\frac{4}{3 + \sqrt{5}} \cdot \frac{3 - \sqrt{5}}{3 - \sqrt{5}}.
  2. 2
    Step 2: Denominator: (3)2−(5)2=9−5=4(3)^2 - (\sqrt{5})^2 = 9 - 5 = 4.
  3. 3
    Step 3: 4(3−5)4=3−5\frac{4(3 - \sqrt{5})}{4} = 3 - \sqrt{5}.
  4. 4
    Check: 43+2.236≈0.764\frac{4}{3 + 2.236} \approx 0.764 and 3−2.236=0.7643 - 2.236 = 0.764 ✓

Answer

3−53 - \sqrt{5}
When the denominator is a binomial containing a radical, multiply by its conjugate. The conjugate of a+ba + \sqrt{b} is a−ba - \sqrt{b}. The product eliminates the radical via difference of squares.

About Rationalizing Denominators

The process of eliminating radical expressions from the denominator of a fraction by multiplying the numerator and denominator by an appropriate expression (the radical itself or its conjugate).

Learn more about Rationalizing Denominators →

More Rationalizing Denominators Examples

Example 1 easy

Rationalize the denominator of [formula].

Example 3 easy

Rationalize [formula].

Example 4 medium

Rationalize [formula].

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