By Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Dave Sobecki

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Combine like terms. Rationalize the denominator and write the answer in simplified radical form. (A) 6 1 Ϫ 13 (B) 21x Ϫ 31y 1x ϩ 1y ANSWERS TO MATCHED PROBLEMS 1. 000 01 (C) x4 (D) v3րu7 3 4 2. (A) 15x (B) 3ր(2y ) (C) Ϫ14x (D) y6 ր(9x8) 4 8 Ϫ3 3. 000 007 42 4. 11 ϫ 10Ϫ19 5. (A) 2 (B) Not real (C) 10 (D) Ϫ1 (E) Ϫ3 6. (A) Ϫ32 (B) 16 (C) 10y13ր12 (D) 2 րx1ր18 4 3 x 15xy 152 x 3 7. (A) 3x2y 12y (B) 2 (C) (D) 2x2y x y 2x Ϫ 51xy ϩ 3y 8. (A) Ϫ3 Ϫ 313 (B) xϪy R-2 (F) 0 Exercises All variables represent positive real numbers and are restricted to prevent division by 0.

3x2 Ϫ 6x) ϩ (4x Ϫ 8) Remove common factors from each group. ϭ 3x(x Ϫ 2) ϩ 4(x Ϫ 2) Factor out the common factor (x ؊ 2). ϭ (3x ϩ 4)(x Ϫ 2) (B) wy ϩ wz Ϫ 2xy Ϫ 2xz Group the first two and last two terms—be careful of signs. ϭ (wy ϩ wz) Ϫ (2xy ϩ 2xz) Remove common factors from each group. ϭ w( y ϩ z) Ϫ 2x(y ϩ z) Factor out the common factor ( y ؉ z). ϭ (w Ϫ 2x)(y ϩ z) (C) 3ac ϩ bd Ϫ 3ad Ϫ bc In parts (A) and (B) the polynomials are arranged in such a way that grouping the first two terms and the last two terms leads to common factors.

B) Explain why the prime number 2 appears an even number of times (possibly 0 times) as a factor in the prime factorization of a2. (C) Explain why the prime number 2 appears an odd number of times as a factor in the prime factorization of 2b2. (D) Explain why parts (B) and (C) contradict the fundamental theorem of arithmetic. qxd 7/14/09 8:39 PM Page 31 SECTION R–3 90. To show that 1n is an irrational number unless n is a perfect square, explain how the assumption that 1n is rational leads to a contradiction of the fundamental theorem of arithmetic by the following steps: (A) Assume that n is not a perfect square, that is, does not belong to the sequence 1, 4, 9, 16, 25, .