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Rationalizing

Rationalizing. MATH 017 Intermediate Algebra S. Rook. Overview. Section 7.5 in the textbook Rationalizing a denominator with one term Rationalizing a denominator with two terms. Rationalizing Denominators with One Term. Rationalizing Denominators with One Term.

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Rationalizing

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  1. 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

  3. Rationalizing Denominators with One Term

  4. Rationalizing Denominators with One Term • 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:

  13. 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 • 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

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