Rationalizing a Monomial Denominator
When a denominator is a single term with a square root, the goal is to rewrite the fraction so no radical remains in the denominator. For a monomial denominator, this is done by multiplying the numerator and denominator by the same factor that makes the denominator rational.
Method
- Identify the radical in the denominator.
Look for a factor like (\sqrt{a}), (\sqrt[n]{a}), or a coefficient times a radical.
- Choose the missing factor.
Pick the factor that turns the denominator into a perfect power or a whole number. For example, (\frac{1}{\sqrt{5}}) needs another (\sqrt{5}), because (\sqrt{5}\cdot\sqrt{5}=5).
- Multiply top and bottom by that factor.
This keeps the value of the fraction the same.
- Simplify completely.
Reduce coefficients, combine radicals if needed, and write the denominator without radicals.
Example idea
If the denominator is (3\sqrt{2}), multiply by (\sqrt{2}). Then the denominator becomes (3\cdot 2=6).
Check your work
Make sure the final denominator contains no radical and that the fraction is equivalent to the original. If possible, simplify any square roots in the numerator as well.