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: Multiply top and bottom by the radical or its conjugate so the denominator becomes rational.

Common stuck point: The procedure for rationalizing denominators is the easy part; the trap is multiplying only the denominator. Asking "Is there a square root in the denominator, and is it a single term or a binomial?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is there a square root in the denominator, and is it a single term or a binomial?

Worked Examples

Example 1

easy

Rationalize the denominator of

\frac{5}{\sqrt{3}}

Answer

\frac{5\sqrt{3}}{3}

First step

Step 1: Multiply numerator and denominator by

\sqrt{3}

Full solution

2
Step 2: $\frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$ .
3
Check: $\frac{5}{\sqrt{3}} \approx \frac{5}{1.732} \approx 2.887$ and $\frac{5(1.732)}{3} \approx 2.887$ ✓

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}}

Example 3

medium

Rationalize

\frac{1}{\sqrt{3} + 1}

Example 4

hard

Rationalize

\frac{\sqrt{2} + 1}{\sqrt{2} - 1}

Example 5

challenge

Rationalize

\frac{1}{1 + \sqrt{2} + \sqrt{3}}

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}

Example 3

easy

Rationalize

\frac{1}{\sqrt{2}}

Example 4

easy

Rationalize

\frac{3}{\sqrt{5}}

Example 5

easy

Rationalize

\frac{2}{\sqrt{3}}

Example 6

easy

Rationalize

\frac{5}{\sqrt{7}}

Example 7

easy

Rationalize

\frac{1}{\sqrt{6}}

Example 8

easy

Rationalize

\frac{4}{\sqrt{2}}

and simplify.

Example 9

easy

Rationalize

\frac{6}{\sqrt{3}}

and simplify.

Example 10

easy

Rationalize

\frac{\sqrt{2}}{\sqrt{5}}

Example 11

medium

Rationalize

\frac{1}{3 + \sqrt{2}}

Example 12

medium

Rationalize

\frac{2}{\sqrt{5} - 1}

Example 13

medium

Rationalize

\frac{\sqrt{3}}{\sqrt{3} + 2}

Example 14

medium

Rationalize

\frac{4}{\sqrt{6} + \sqrt{2}}

Example 15

medium

Rationalize

\frac{5}{\sqrt{8}}

and simplify.

Example 16

medium

Rationalize

\frac{1}{\sqrt{x}}

for

x>0

Example 17

medium

Rationalize

\frac{3}{2 - \sqrt{3}}

Example 18

medium

Rationalize

\frac{1}{4 + \sqrt{5}}

Example 19

medium

Rationalize

\frac{6}{\sqrt{12}}

and simplify.

Example 20

challenge

Rationalize

\frac{\sqrt{5} + 1}{\sqrt{5} - 1}

Example 21

challenge

Rationalize

\frac{1}{\sqrt{x} - \sqrt{y}}

for

x\neq y

x,y>0

Example 22

challenge

Rationalize

\frac{2}{\sqrt{7} + \sqrt{3}}

and simplify.

Example 23

easy

Rationalize

\frac{4}{\sqrt{5}}

Example 24

easy

Rationalize

\frac{7}{\sqrt{11}}

Example 25

easy

Rationalize

\frac{8}{\sqrt{2}}

and simplify.

Example 26

medium

Rationalize

\frac{3}{2\sqrt{5}}

Example 27

medium

Rationalize

\frac{5}{\sqrt{7} - \sqrt{2}}

Example 28

medium

Rationalize

\frac{2}{\sqrt{3} + \sqrt{5}}

Example 29

medium

Rationalize

\frac{\sqrt{3}}{\sqrt{6}}

Example 30

medium

Rationalize

\frac{1}{\sqrt{2} + \sqrt{3}}

Example 31

medium

Rationalize

\frac{2}{5 - \sqrt{3}}

Example 32

medium

Rationalize

\frac{3}{\sqrt{x}}

for

x > 0

Example 33

hard

Rationalize

\frac{\sqrt{5}}{\sqrt{5} + 2}

Example 34

hard

Rationalize

\frac{2}{\sqrt[3]{4}}

Example 35

hard

Rationalize

\frac{6}{\sqrt{8}}

and simplify.

Example 36

hard

Rationalize

\frac{1}{\sqrt{x} + \sqrt{y}}

for

x, y > 0

x \neq y

Example 37

hard

Rationalize

\frac{\sqrt{3} - 1}{\sqrt{3} + 1}

Example 38

medium

Rationalize

\frac{10}{\sqrt{50}}

and simplify.

Example 39

hard

Rationalize

\frac{4}{\sqrt{6} - 2}

Example 40

hard

Rationalize

\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}

Example 41

medium

Rationalize

\frac{5}{2\sqrt{3} + 1}

Example 42

challenge

Rationalize

\frac{1}{\sqrt{x+h} - \sqrt{x}}

(used in calculus derivative-of-

\sqrt{x}

derivation).

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Related Concepts

Simplifying Radicals Division Radical Equations

Background Knowledge

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

simplifying radicalsdivision