254 CHAPTER 5 Number Theory and the Real Number System Rationalizing the Denominator A denominator of a fraction is rationalized when it contains no radical expressions. To rationalize a denominator that contains only a square root, multiply both the numerator and denominator of the fraction by the square root of a number that will result in the radicand in the denominator becoming a perfect square. This action is the equivalent of multiplying the fraction by 1 because the value of the fraction does not change. Then simplify the fractions when possible. Did You Know? Why We Rationalize the Denominator Historically, denominators were rationalized so that obtaining a decimal approximation would be easier. For example, by rationalizing the denominator, we can see that 1 5 5 5 . = Approximating 1 5 using long division requires you to divide 1 by an approximation of 5 such as 2.236. However, approximating 5 5 requires you to divide 2.236 by 5 which is a much easier task! Example 6 Rationalizing the Denominator Rationalize the denominator of the following. a) 5 2 b) 5 12 c) 5 10 Solution a) Multiply the numerator and denominator by the square root of a number that will make the radicand in the denominator a perfect square. 5 2 5 2 2 2 5 2 4 5 2 2 = ⋅ = = Note that the 2’s in the answer cannot be divided out because one 2 is a radicand and the other is not. b) 5 12 5 12 3 3 5 3 36 5 3 6 = ⋅ = = You could have also obtained the same answer to this problem by multiplying both the numerator and denominator by 12 and then simplifying. c) Write 5 10 as 5 10 and reduce the fraction to obtain 1 2 . By the quotient rule for radicals, 1 2 = 1 2 or 1 2 . Now rationalize the denominator of 1 2 . 1 2 1 2 2 2 2 2 = ⋅ = The answer could also be obtained by multiplying the numerator and denominator by 10 and then simplifying. 7 Now try Exercise 61 Estimating Square Roots Without a Calculator Consider the irrational number 17. The radicand 17 is between the two perfect squares 16 and 25. Since 16 4 = and 25 5, = we can reason that 17 is between 4 and 5. We summarize this discussion as follows. 16 17 25 16 17 25 4 17 5 < < < < < < Furthermore, since 17 is closer to 16 than it is to 25, we can further estimate that 17 is closer to 4 than it is to 5. In other words, 17 is between 4 and 4.5.
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