Rationalize a One-Term Denominator
Rationalizing a one-term denominator eliminates a square root from the bottom of a fraction by multiplying numerator and denominator by that same square root. From OpenStax Elementary Algebra 2E, Chapter 9, for 5/sqrt(2): multiply by sqrt(2)/sqrt(2) to get 5*sqrt(2)/2. For sqrt(3)/sqrt(6): multiply by sqrt(6)/sqrt(6) to get sqrt(18)/6 = 3*sqrt(2)/6 = sqrt(2)/2. A denominator is rationalized when it contains no radical. The technique uses the fact that sqrt(a)*sqrt(a)=a.
Key Concepts
Property Rationalizing the Denominator The process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer is called rationalizing the denominator .
Simplified Square Roots A square root is considered simplified if there are no perfect square factors in the radicand no fractions in the radicand no square roots in the denominator of a fraction.
Examples To rationalize $\frac{5}{\sqrt{2}}$, multiply the top and bottom by $\sqrt{2}$: $\frac{5 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{5\sqrt{2}}{2}$.
Common Questions
How do you rationalize a one-term denominator?
Multiply both the numerator and denominator by the same square root that appears in the denominator. This makes the denominator sqrt(a)×sqrt(a)=a, eliminating the radical.
Rationalize 5/sqrt(2).
Multiply by sqrt(2)/sqrt(2): (5·sqrt(2))/(sqrt(2)·sqrt(2)) = 5*sqrt(2)/2.
Rationalize sqrt(3)/sqrt(6).
Multiply by sqrt(6)/sqrt(6): sqrt(18)/6. Simplify sqrt(18)=3*sqrt(2). Result: 3*sqrt(2)/6 = sqrt(2)/2.
What does it mean to have a rationalized denominator?
A rationalized denominator has no radical (square root) in it. The denominator is a rational number. This is the standard simplified form for radical fractions.
Why is rationalizing the denominator required in simplified form?
Standard form of a fraction does not include radicals in the denominator. Rationalized form is easier to compare, estimate, and use in further calculations.