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Rationalizing the Denominator

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

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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!!!

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Example 1 Simplify by rationalizing the denominator:

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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?

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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

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Example 3 Simplify by rationalizing the denominator:

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Example 4 Simplify by rationalizing the denominator:

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