Algebra 2 · Unit 5 · Topic 5.3
Radical Arithmetic and Rationalizing Denominators
Like radicals can be added by combining their coefficients after simplification.
Definition and explanation
Like radicals can be added by combining their coefficients after simplification. Products of real square roots satisfy √a√b = √(ab) for a, b ≥ 0. Rationalizing rewrites a denominator without a radical using an equivalent nonzero factor.
Worked example
Simplify 3√12 − √27: it becomes 6√3 − 3√3 = 3√3. To rationalize 1/(√5 + 1), multiply by (√5 − 1)/(√5 − 1), obtaining (√5 − 1)/4.
Common mistake
Unlike radicals cannot generally be combined into a single radical by adding their radicands.
Key takeaway
Like radicals can be added by combining their coefficients after simplification.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.