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Rationalizing the Denominator

Learn how to get radicals out of the denominator of a fraction using rationalization. Understand the process and strategies for simplifying fractions with square root denominators effectively. Includes examples and tips for better understanding.

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Rationalizing the Denominator

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  1. Rationalizing the Denominator

  2. Essential Question • How do I get a radical out of the denominator of a fraction?

  3. If a fraction has a denominator that is a square root, you can simplify it by rationalizing the denominator. • To do this: Multiply both the numerator and the denominator by a number that produces a perfect square under the radical sign in the denominator. • TIP: Always look to SIMPLIFY at the beginning of the problem. This usually makes the simplifying or reducing at the end easier!! • WE DON’T LEAVE SQUARE ROOTS IN THE DENOMINATOR!!!

  4. Example 1 • Simplify by rationalizing the denominator:

  5. Example 2 • Simplify by rationalizing the denominator: Is there a simpler way of doing this problem? Look at the beginning. Can we simplify the original expression?

  6. PART II: Using a CONJUGATE to rationalize a binomial denominator • A conjugate is… a binomial with the opposite operation. • Number pairs in the form a + √(b) and a – √(b) are conjugates. • Example: The conjugate of 2 + √(3) is 2 – √(3) • Example: The conjugate of –3 – 4 √(5) is –3 + 4√(5) • Multiplying a number by its conjugate produces a rational number. • Example: (2 + √(3)) (2 – √(3)) = 4 – 2√(3) + 2√(3) – 3 = 1 • You try: (-1 + 2√(5)) (-1 – 2√(5)) • = 1 + 2√(5) – 2√(5) – 4(5) = 1 – 20 = –19

  7. Example 3 • Simplify by rationalizing the denominator:

  8. Example 4 • Simplify by rationalizing the denominator:

  9. Simplify by rationalizing the denominator:

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