By I. N. Herstein
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51͞2 ϭ Ί5 Rewrite as radical. Radical expressions can be combined (added or subtracted) if they are like radicals—that is, if they have the same index and radicand. For instance, 2Ί3x 3 and 12Ί3x are like radicals, but Ί 3x and 2Ί3x are not like radicals. Example 8 Simplifying and Combining Like Radicals a. 2Ί48 ϩ 3Ί27 ϭ 2Ί16 b. 3 16x Ί Ϫ 3 54x Ί и 3 ϩ 3Ί9 и 3 Find square factors. ϭ 8Ί3 ϩ 9Ί3 Find square roots. ϭ 17Ί3 Combine like terms. и 2x ϭ 3 8 Ί ϭ 3 2x 2Ί 3 27 ϪΊ Ϫ и 2x Find cube factors.
To rationalize the denominator of the following fraction, multiply both the numerator and the denominator by Ί3 to obtain 5 2Ί3 ✓CHECKPOINT 4 1 by rationalizing the 4 denominator. ■ Simplify 3 Ί ϭ Ί3 5 2Ί3 иΊ ϭ 3 5Ί3 5Ί3 5Ί3 ϭ ϭ . 2 2Ί3 2͑3͒ 6 b. To rationalize the denominator of the following fraction, multiply both the 3 52. Note how this eliminates the radical numerator and the denominator by Ί from the denominator by producing a perfect cube in the radicand. 2 2 ϭ 3 3 Ί5 Ί5 Example 5 3 52 Ί иΊ 3 52 ϭ 3 3 52 2Ί 25 2Ί ϭ 3 3 5 Ί5 Rationalizing a Denominator with Two Terms 2 2 ϭ Ί 3 ϩ 7 3 ϩ Ί7 ϭ и 3 Ϫ Ί7 3 Ϫ Ί7 2͑3 Ϫ Ί7 ͒ 32 Ϫ ͑Ί7͒2 Multiply numerator and denominator by conjugate.
This is called an inclusive or, and it is the way the word o“ r”is always used in mathematics. Properties of Fractions Let a, b, c, and d be real numbers, variables, or algebraic expressions such that b 0 and d 0. Then the following properties are true. c a 1. Equivalent fractions: ϭ if and only if ad ϭ bc. b d a Ϫa a 2. Rules of signs: Ϫ ϭ ϭ b b Ϫb 3. Generate equivalent fractions: and Ϫa a ϭ Ϫb b a ac ϭ , c b bc 4. Add or subtract with like denominators: a±c a c ± ϭ b b b 5. Add or subtract with unlike denominators: 6.