There are multiple concepts being tested on this question and that’s probably why this has been categorized as a 700-level question. Most of these concepts though, are concepts on exponents.
\(X^0\) = 1 provided x≠0, is one of the concepts being tested here.
\(X^{-n}\) = \(\frac{1}{X^n}\) is another concept that we will use in evaluating the statements.
Lastly, the concept that the absolute value of a number is always positive is something that will help you evaluate statement I and statement III.
We know that a=-1. So, the base is a negative number. When the base is negative, you need to be more careful because the final value of the number depends on the power also.
If the base is a negative value and the exponent is odd, then the resultant value will be negative. If the base is a negative value and the exponent is even, the resultant value will be positive. Note that I did not specify that the base is a negative INTEGER. So, what is generalized above applies to all negative real numbers, regardless of they being integers or otherwise. Also note that odd and even is defined for negative numbers as well.
Evaluating statement I, which is \(a^a\) = -|a|, we see that \(a^a\) = \((-1)^{-1}\) which simplifies to \(\frac{1}{(-1)^1}\) leading to \(\frac{1}{(-1)}\). This means, the LHS = -1.
|a| = |-1| = 1. Therefore, -|a| = -1, which is the RHS.
Clearly, LHS = RHS. Statement I is true. And because of this, options B and C can be eliminated. The possible answer options at this stage are A, D or E.
The LHS in statement II is equal to 1, since any non-zero value raised to the power of ZERO is 1. Observe that the RHS has \(-a^{-1}\). This means \(–(-1)^{-1}\) which works out to 1. LHS = RHS, statement II is true.
The LHS in statement III is positive because it represents the absolute value of a, whereas the RHS of statement III is negative. Statement III is not true.
Answer options A and E can be eliminated, the correct answer option is D.
Hope that helps!
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