Preparing for University Calculus - Math and Computer Science

3.7 Rationalizing numerators or denominatorsYou should know how to eliminate square (and other) roots from the numerator ordenominator of a fraction by multiplying both the numerator and denominator byan appropriate expression. This technique will be important in finding the derivativesof certain expressions involving roots.Examples:11. Rationalize the denominator of √ aSolution: By multiplying both numerator and denominator by √ a weobtain1√ a= 1 √ a√ a√ a=2. Rationalize the denominator of 1+√ x2+ √ xSolution: Recall, when we multiply (a + b)(a − b) we obtain a differenceof squares a 2 − b 2 . So, if the denominator of a rational expression containsa constant added to a square root term, we can eliminate the root term bymultiplying by the difference of the constant and the root term. In this caseif we were to multiply both the numerator and the denominator by 2 − √ xwe will obtain the desired result.1+ √ x2+ √ x = (1 + √ x)(2 + √ (2 − √ x)x) (2 − √ x)= 2 − √ x +2 √ x − x4 − 2 √ x +2 √ x − x= 2+√ x − x4 − x3. Rationalize the numerator of 1+√ x2+ √ xSolution: This time we multiply numerator and denominator by theconjugate of the numerator:1+ √ x2+ √ x = (1 + √ x)(2 + √ (1 − √ x)x) (1 − √ x) = 1 − x2 − √ x − x22√ aa