Rationalizing the Denominator
Rationalizing the Denominator is the process of rewriting a fraction so that no radical appears in the denominator, taught in Yoshiwara Elementary Algebra Chapter 9: More About Exponents and Roots. Grade 6 students multiply both numerator and denominator by the radical in the denominator, turning it into a perfect square and eliminating the radical. This technique produces a simplified, conventional form for expressions containing square roots.
Key Concepts
Property The process of eliminating a radical from the denominator of a fraction is called rationalizing the denominator . This is done by multiplying the numerator and the denominator by the same root that appeared in the denominator originally. It is often best to simplify any radicals in the expression before attempting to rationalize.
Examples To rationalize $\frac{7}{\sqrt{3}}$, multiply the numerator and denominator by $\sqrt{3}$: $\frac{7 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{7\sqrt{3}}{3}$.
To rationalize $\frac{6}{\sqrt{12}}$, first simplify the denominator: $\frac{6}{2\sqrt{3}} = \frac{3}{\sqrt{3}}$. Now rationalize: $\frac{3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{3\sqrt{3}}{3} = \sqrt{3}$.
Common Questions
What does rationalizing the denominator mean?
It means rewriting a fraction so the denominator contains no radicals. You multiply both numerator and denominator by the radical term in the denominator.
How do you rationalize a denominator with a square root?
Multiply the top and bottom by the same radical. For example, 1/√3 becomes √3/(√3 × √3) = √3/3.
Why do we rationalize the denominator?
It is the standard simplified form for fractions. Radicals in denominators make expressions harder to compare and add; rationalizing makes computation cleaner.
Where is rationalizing the denominator in Yoshiwara Elementary Algebra?
It is covered in Chapter 9: More About Exponents and Roots of Yoshiwara Elementary Algebra.
What if the denominator has a binomial with a radical?
Multiply by the conjugate (changing the sign between terms). For example, multiply by (a - √b) when the denominator is (a + √b).