Pre-Algebra:
Rationalizing Denominators

1. 
\(\frac{2}{\sqrt{3}}\)
 Solution
2. 
\(\sqrt{\frac{25}{3}}\)
 Solution
3. 
\(\frac{6\sqrt{2}}{\sqrt{3}}\)
 Solution
4. 
\(\frac{3}{2+\sqrt{2}}\)
 Solution
5. 
\(\frac{1+\sqrt{7}}{2-\sqrt{7}}\)
 Solution
6. 
\(\frac{\sqrt{7}+1}{-\sqrt{7}+2}\)
 Solution
7. 
\(\frac{1}{\sqrt{3}}+\frac{3}{\sqrt{5}}\)
 Solution
8. 
\(\frac{3\sqrt{2}}{\sqrt{6}-\sqrt{3}}-\frac{3}{3-\sqrt{6}}\)
 Solution
9. 
\(\frac{-6}{5\sqrt{3}}\)
 Solution
10. 
\(\frac{1}{5\sqrt{3}\sqrt{5}}\)
 Solution
11. 
\(\frac{\sqrt{10}-\sqrt{3}}{\sqrt{10}+\sqrt{3}}\)
 Solution
12. 
\(\frac{\sqrt{3}}{\sqrt{7+\sqrt{5}}}\)
 Solution

Rationalizing Denominators - Introduction

When a radical contains an expression that is not a perfect root, for example, the square root of 3 or cube root of 5, it is called an irrational number. So, in order to rationalize the denominator, we need to eliminate the radicals that are in the denominator.

Tips on Rationalizing Denominators

1. Multiply numerator and denominator by a radical that will eliminate of the radical in the denominator.
If the radical in the denominator is a square root, then you multiply by a square root that will give you a perfect square under the radical when multiplied by the denominator.
2. Make sure all radicals are simplified.
Some radicals could already be in simplified forms, but make sure you simplify the ones that are not.
3. Simplify the fraction if needed.
Be careful. You cannot cancel out a factor that is on the outside of a radical with one that is on the inside of the radical. In order to cancel out common factors, they have to be both inside the same radical or be both outside the radical.

Example

Rationalize the Denominator:
\(\frac{\sqrt{7}}{\sqrt{3}}\)
To rationalize the denominator, you need to multiply both the numerator and denominator by the radical found in the denominator.
\(\frac{\sqrt{7}}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}\)
Multiply both the numerator and the denominator. Remember that you can only multiply numbers outside the radical with numbers outside the radical, and numbers inside the radical with numbers inside the radical.
\(\frac{\sqrt{21}}{\sqrt{9}}\)
Simplify the radicals.
\(\frac{\sqrt{21}}{3}\)
Reduce the fraction if needed. In this case, we cannot reduce the expression any further, so we are done. Our final answer is:
\(\frac{\sqrt{21}}{3}\)

What's Next

Ready to learn more about rationalizing denominators? See our practice problems at the top of this page. You can also visit Cymath’s free online practice problems or upgrade to Cymath Plus. Cymath Plus provides an ad-free experience complete with even more guidance to help jump start your math skills.