By Azriel Rosenfeld

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N 0 ϭ 0 Ί 5 0 ϭ 0 Ί Integers such as 1, 4, 9, 16, 25, and 36 are called perfect squares because they have integer square roots. Similarly, integers such as 1, 8, 27, 64, and 125 are called perfect cubes because they have integer cube roots. Properties of Radicals Let a and b be real numbers, variables, or algebraic expressions such that the indicated roots are real numbers, and let m and n be positive integers. Property n am ϭ Ί 1. Ί ͑ n a ͒m n a 2. Ί 3. 4. n a Ί n b Ί n b ϭΊ n ab иΊ ϭ Ίab , n Ί5 b и Ί7 ϭ Ί5 и 7 ϭ Ί35 4 27 Ί 0 4 9 Ί m n Ίa ϭ mn Ίa Ί ϭ Ί279 ϭ 4 4 3 Ί 3 Ί 6 10 Ί 10 ϭ Ί n a 5.

120. The sum of two binomials is always a binomial. 121. Find the degree of the product of two polynomials of degrees m and n. 122. Find the degree of the sum of two polynomials of degrees m and n if m < n. 123. WRITING A student’s homework paper included the following. ͑x Ϫ 3͒2 ϭ x 2 ϩ 9 Write a paragraph fully explaining the error and give the correct method for squaring a binomial. 124. CAPSTONE A third-degree polynomial and a fourth-degree polynomial are added. (a) Can the sum be a fourth-degree polynomial?

2x 2 ϩ 9x ϩ c 147. GEOMETRY The volume V of concrete used to make the cylindrical concrete storage tank shown in the figure is V ϭ R 2h Ϫ r 2h, where R is the outside radius, r is the inside radius, and h is the height of the storage tank. R h (c) An 80-pound bag of concrete mix yields 35 cubic foot of concrete. Find the number of bags required to construct a concrete storage tank having the following dimensions. Outside radius, R ϭ 4 feet Inside radius, r ϭ 323 feet Height, h feet (d) Use the table feature of a graphing utility to create a table showing the number of bags of concrete required to construct the storage tank in part (c) with heights of h ϭ 12, h ϭ 1, h ϭ 32, h ϭ 2, .

### An introduction to algebraic structures by Azriel Rosenfeld

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