If a = -1, which of the following is (are) true?

Question

If  a = -1 , which of the following is (are) true?

I.  a^a = -|a|

II.  a^0 = -a^{-1}

III.  |a|=a

A. I only 
B. II only 
C. III only 
D. I & II only 
E. I, II & III

abhimahna · Answer

Simple question.

a=-2

a^a = (-1)^-1 = 1/-1 = -1

|a| = |-1|=1

Now, I. a^a=-|a| is correct as per above calculated values.

So, we are left with options A,D and E

now look for statement 2 ,
a^0=-a^-1

a^0 = 1 (always)

a^-1 = -1

So, B is also correct. Options left D and E

since a is negative , |a| <> a.

Hence, answer is D.

paidlukkha · Answer

a=-|a| -1 = -|-1| -1= -*1 -1=-1 so I yes a^0=a^-1 1=1/1 1=1 sp II yes |a|<>a, as |-1|=1 & -1 <>1 so 3rd option out D (I & II only)

MathRevolution · Answer

From (-1)0=1, (-1)-1=-1, |-1|=1, the correct answer is D.

Ansh777 · Answer

Given: a=-1

I. a^a=-|a|  
LHS: (-1)^(-1)=-1
RHS: -|-1|=-1
Therefore a^a=-|a|

II. a^0=-a^-1  
LHS: (-1)^0=1
RHS: -((-1)^(-1))=1
Therefore a^0=-a^-1

III. |a|=a 
LHS:|-1|=1
RHS: -1
Therefore |a| is not equal to a

Answer: D

CrackverbalGMAT · Answer

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!

sasyaharry · Answer

How can one complete such problems in less than 1:30? I took 1:44 and got the correct answer but feel like this is too much time.

EgmatQuantExpert · Answer

Solution

Given  
In this question, we are given that
 • The value of the variable a = -1

To find  
We need to determine
 • From the given choices, which one is true

Approach and Working out  
Let us proceed with simplifying each of the options separately
 I. a^a = (-1)^(-1) = 1/(-1) = -1 and -|a| = -|-1| = -1 Hence, I is true
II. a^0 = 1 and -a^-1 = -(1/a) = -(1/-1) = -(-1) = 1 Hence, II is true
III. |a| = |-1| = 1 and a = -1 Hence, III is not true

Therefore, from the given choices, I and II are true whereas III is not true.
Thus, option D is the correct answer.

Correct Answer: Option D

AnushkaTanwar · Answer

Can anyone tell me why are using only one value in option A, ain't there two?

Bunuel · Answer

We are given that a = -1. Which other value do you want to check and why? Also, which other value do you suggest satisfies a^a = -|a|?

AnushkaTanwar · Answer

I might be wrong but i want a clearance, so the doubt is:
If, a=1, then there will be two value, |a|=1 or -1, right 
then why not the same in the case of a=-1, -|a|=-|-1|, this will come as one positive value and the other as same right?,
i am confused in this only, nothing else, if -1 from the modulus will come as only positive and not as -1

Bunuel · Answer

Not quite sure what you mean but |-1| = 1, only.

10. Absolute Value 
 
    Theory    
 
 Math: Absolute value (Modulus) 
 Absolute Value: Tips and hints 
 Properties of Absolute Values on the GMAT 
 Absolute modulus : A better understanding 
 GMAT Question Patterns: Abolute Value 
 The Reason Behind Absolute Value Questions on the GMAT 
 
    Questions    
 
 Inequality and absolute value questions from my collection 
 The E-GMAT Question Series on ABSOLUTE VALUE 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

If a = -1, which of the following is (are) true?

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If a = -1, which of the following is (are) true?

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