SOLUTION: Which of the following products is a rational number? 1: (2+sqrt[2])(2+sqrt[2]) 2: (2+sqrt[2])(2-sqrt[2]) 3: sqrt[2](2+sqrt[2]) 4: 2(2-sqrt[2])

Algebra ->  Test -> SOLUTION: Which of the following products is a rational number? 1: (2+sqrt[2])(2+sqrt[2]) 2: (2+sqrt[2])(2-sqrt[2]) 3: sqrt[2](2+sqrt[2]) 4: 2(2-sqrt[2])      Log On


   

Question 389113: Which of the following products is a rational number?
1: (2+sqrt[2])(2+sqrt[2])
2: (2+sqrt[2])(2-sqrt[2])
3: sqrt[2](2+sqrt[2])
4: 2(2-sqrt[2])
Found 2 solutions by richard1234, haileytucki:
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
#2 is rational, it is a difference of two squares problem.

%282+%2B+sqrt%282%29%29%282+-+sqrt%282%29%29+=+2%5E2+-+%28sqrt%282%29%29%5E2+=+2

Answer by haileytucki(390) About Me  (Show Source):
You can put this solution on YOUR website!
#2 (2+~(2))(2-~(2))
Multiply each term in the first group by each term in the second group using the FOIL method. FOIL stands for First Outer Inner Last, and is a method of multiplying two binomials. First, multiply the first two terms in each binomial group. Next, multiply the outer terms in each group, followed by the inner terms. Finally, multiply the last two terms in each group.
(2*2+2*-~(2)+~(2)*2+~(2)*-~(2))
Simplify the FOIL expression by multiplying and combining all like terms.
(4-~(2)^(2))
Remove the parentheses around the expression 4-~(2)^(2).
4-~(2)^(2)
Raising a square root to the square power results in the expression inside the root.
4-(2)
Multiply -1 by the 2 inside the parentheses.
4-2
Subtract 2 from 4 to get 2.
2