Official Explanation
This is a Yes/No DS question. Answering a definite
"Yes" or a definite
"No" means Sufficient. If the answer is sometimes
"Yes" and sometimes
"No," it means Maybe, which means Insufficient.
The issue is whether \(|a|+|b|\) is even. Since the rules of even and odd are the same for addition and subtraction, positive vs. negative, the real issue of the question is whether \(a+b\) is even. If \(a+b\) is even, then \(|a|+|b|\) will also be even, and vice versa.
Plug in numbers that satisfy the statements and show that the expression is even and the answer is
"Yes." Then ask yourself,
"Is this always true, for any number?" Think of DOZEN F numbers to reach an answer of
"No" and prove the statement(s) Insufficient.
Stat. (1): multiply the powers with the same base of \(x: x^a⋅x^b=x{a+b}=1.\) This seems to indicate that \(a+b=0\), since \(x^0=1.\) Since zero is even, then answer is
"Yes," But is it always
"Yes" for any number? If \(x=1\), then \(a+b\) could be anything (for instance, \(1^3=1\) also). Therefore, if \(x=1\) the answer could be
"No," That's a
"Maybe," so
Stat.(1) → IS → BCE.Stat. (2) alone tells you nothing about \(b\). If \(a=−2\) and \(b=0\), then \(|a|+|b|=|−2|+|0|=2\), which is even, yielding an answer of
"Yes." However, if \(a=−2\) and \(b=1\), then \(|a|+|b|=|−2|+|1|=3\), which is odd, yielding an answer of
"No." No definite answer, so
Stat.(2) → IS → CE.Stat. (1+2) combined still allow both cases—either \(|a|+|b|=0\) and the answer is
"Yes," or \(x=1\), which then allows \(|a|+|b|\) to be an odd number, yielding a
"No" answer. Still no definite answer, so
Stat.(1+2) → IS → E.Answer: E _________________