Rationalizing MATH 017 Intermediate Algebra S. Rook

2 Overview Section 7.5 in the textbook –Rationalizing a denominator with one term –Rationalizing a denominator with two terms

Rationalizing Denominators with One Term

4 Rationalizing: the process of eliminating the radical from either the numerator or denominator of a fraction –We will only be rationalizing the denominator Consider – what happens when we multiply it by itself?

5 Rationalizing Denominators with One Term (Continued) Thus, we can say: To rationalize a denominator with one term: –Determine what needs to be multiplied to eliminate the radical in the denominator –Multiply this term times BOTH the numerator and denominator (dealing with an expression) This eliminates the radical in the denominator –Acceptable to have radicals present in the numerator

6 Rationalizing Denominators with One Term (Example) Ex 1: Rationalize the denominator:

7 Rationalizing Denominators with One Term (Example) Ex 2: Rationalize the denominator:

8 Rationalizing Denominators with One Term (Example) Ex 3: Rationalize the denominator:

9 Simplify Radicals Before Rationalizing Consider rationalizing –Could multiply numerator and denominator by –Easier, however to simplify Multiply numerator and denominator by See if radicals can be simplified before rationalizing –Otherwise radicals must be simplified at the end where the numbers are larger

10 Simplify Radicals Before Rationalizing (Example) Ex 4: Rationalize the denominator:

11 Simplify Radicals Before Rationalizing (Example) Ex 5: Rationalize the denominator:

12 Simplify Radicals Before Rationalizing (Example) Ex 6: Rationalize the denominator:

Rationalizing Denominators with Two Terms

14 Rationalizing Denominators with Two Terms Again, goal is to eliminate all radicals from the denominator –Consider (x + 2)(x – 2) –Consider For the last 2, notice all terms with radicals add out

15 Rationalizing Denominators with Two Terms (Continued) Conjugate: same 2 terms but different sign –Given, what would be its conjugate? To rationalize a denominator with two terms: –Multiply BOTH the numerator and denominator by the conjugate (dealing with an expression) This eliminates the radicals in the denominator –Acceptable to have radicals present in the numerator Look to simplify radicals before rationalizing

16 Simplify Radicals Before Rationalizing (Example) Ex 7: Rationalize the denominator:

17 Simplify Radicals Before Rationalizing (Example) Ex 8: Rationalize the denominator:

18 Simplify Radicals Before Rationalizing (Example) Ex 9: Rationalize the denominator:

19 Simplify Radicals Before Rationalizing (Example) Ex 10: Rationalize the denominator:

20 Summary After studying these slides, you should know how to do the following: –Rationalize denominators containing one or two terms