Practice Rationalizing Denominators in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick 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 aโ‹…a=a\sqrt{a} \cdot \sqrt{a} = a, which eliminates the radical from the bottom. For binomial denominators like 3+23 + \sqrt{2}, multiply by the conjugate 3โˆ’23 - \sqrt{2} to use the difference of squares pattern.

Showing a random 20 of 50 problems.

Example 1

medium
Rationalize 12+3\frac{1}{\sqrt{2} + \sqrt{3}}.

Example 2

challenge
Rationalize 1xโˆ’y\frac{1}{\sqrt{x} - \sqrt{y}} for xโ‰ yx\neq y, x,y>0x,y>0.

Example 3

medium
Rationalize 523+1\frac{5}{2\sqrt{3} + 1}.

Example 4

medium
What is the conjugate of 7โˆ’3\sqrt{7} - 3?

Example 5

challenge
Rationalize 1x+hโˆ’x\frac{1}{\sqrt{x+h} - \sqrt{x}} (used in calculus derivative-of-x\sqrt{x} derivation).

Example 6

easy
Rationalize 57\frac{5}{\sqrt{7}}.

Example 7

hard
Rationalize 46โˆ’2\frac{4}{\sqrt{6} - 2}.

Example 8

medium
Rationalize 57โˆ’2\frac{5}{\sqrt{7} - \sqrt{2}}.

Example 9

medium
Rationalize 3x\frac{3}{\sqrt{x}} for x>0x > 0.

Example 10

easy
Rationalize 23\frac{2}{\sqrt{3}}.

Example 11

easy
Rationalize 27\frac{2}{\sqrt{7}}.

Example 12

medium
Rationalize 46+2\frac{4}{\sqrt{6} + \sqrt{2}}.

Example 13

easy
Rationalize 82\frac{8}{\sqrt{2}} and simplify.

Example 14

challenge
Rationalize 11+2+3\frac{1}{1 + \sqrt{2} + \sqrt{3}}.

Example 15

medium
Rationalize 325\frac{3}{2\sqrt{5}}.

Example 16

medium
Rationalize 33+2\frac{\sqrt{3}}{\sqrt{3} + 2}.

Example 17

medium
Rationalize 58\frac{5}{\sqrt{8}} and simplify.

Example 18

easy
Rationalize 711\frac{7}{\sqrt{11}}.

Example 19

medium
Rationalize 25โˆ’3\frac{2}{5 - \sqrt{3}}.

Example 20

medium
Rationalize 25โˆ’1\frac{2}{\sqrt{5} - 1}.