Rationalizing the Denominator
Rationalize the denominator by eliminating radicals from the bottom of a fraction: multiply by the conjugate or by sqrt(n)/sqrt(n) to produce an equivalent expression with integer denominator.
Key Concepts
To rationalize the denominator of an expression, write an equivalent expression so that there are no radicals in any denominator and no denominators in any radical. An expression is in simplest form when no radicand has a perfect square root factor and there are no irrational radical expressions in any denominator.
Example 1: $\frac{3}{2\sqrt{5}} = \frac{3}{2\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{10}$ Example 2: $\sqrt{\frac{1}{12}} = \sqrt{\frac{1 \cdot 3}{12 \cdot 3}} = \frac{\sqrt{3}}{\sqrt{36}} = \frac{\sqrt{3}}{6}$ Example 3: $\frac{7}{\sqrt{x}} = \frac{7}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}} = \frac{7\sqrt{x}}{x}$.
Think of an irrational denominator like wearing socks with sandals—it's just not done in the world of simplified math! Rationalizing is a style makeover for your fractions. We multiply the top and bottom by a special number to kick that pesky radical out of the denominator, making the expression clean, tidy, and officially in its simplest form.
Common Questions
What does it mean to rationalize the denominator?
Rationalizing the denominator means rewriting a fraction so there are no radicals (square roots) in the denominator. This is done by multiplying the numerator and denominator by an appropriate expression that eliminates the radical from the bottom.
How do you rationalize a denominator with a single radical like 1/sqrt(3)?
Multiply numerator and denominator by sqrt(3): (1/sqrt(3)) * (sqrt(3)/sqrt(3)) = sqrt(3)/3. The denominator becomes 3 because sqrt(3)*sqrt(3) = 3. The result sqrt(3)/3 is the rationalized form.
How do you rationalize a denominator with a binomial like 1/(2+sqrt(5))?
Multiply by the conjugate (2-sqrt(5))/(2-sqrt(5)). The denominator becomes (2+sqrt(5))(2-sqrt(5)) = 4-5 = -1. The numerator becomes 2-sqrt(5). So the result is (2-sqrt(5))/(-1) = sqrt(5)-2.