Rationalizing Denominators Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Rationalizing Denominators.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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).

A radical in the denominator is considered 'messy.' To clean it up, multiply top and bottom by the same radical (or conjugate). This works because \sqrt{a} \cdot \sqrt{a} = a, which eliminates the radical from the bottom. For binomial denominators like 3 + \sqrt{2}, multiply by the conjugate 3 - \sqrt{2} to use the difference of squares pattern.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Multiplying by the conjugate leverages the difference of squares identity to eliminate radicals: (a + \sqrt{b})(a - \sqrt{b}) = a^2 - b.

Common stuck point: For binomial denominators, you MUST use the conjugate, not just the radical. The conjugate of a + \sqrt{b} is a - \sqrt{b}.

Sense of Study hint: Write down the conjugate of the denominator, then multiply both top and bottom by it.

Worked Examples

Example 1

easy
Rationalize the denominator of \frac{5}{\sqrt{3}}.

Solution

  1. 1
    Step 1: Multiply numerator and denominator by \sqrt{3}.
  2. 2
    Step 2: \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}.
  3. 3
    Check: \frac{5}{\sqrt{3}} \approx \frac{5}{1.732} \approx 2.887 and \frac{5(1.732)}{3} \approx 2.887 ✓

Answer

\frac{5\sqrt{3}}{3}
Rationalizing means removing radicals from the denominator. Multiply top and bottom by the radical in the denominator — this creates a perfect square in the denominator.

Example 2

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Rationalize \frac{2}{\sqrt{7}}.

Example 2

medium
Rationalize \frac{6}{\sqrt{2} - 1}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

simplifying radicalsdivision