If x 0, what is the value of x ? (1) |x^x| = 1 (2) x^5 = x^x

Question

If x ≠ 0, what is the value of x ?

(1)  |x^x| = 1

(2)  x^5 = x^x

M36-134

DavidTutorexamPAL · Answer

We'll try finding numbers which give contradictory answers.
This is an Alternative approach.

(1) So x^x must equal 1 or -1. This can happen if the exponent is 0 which is impossible by question definition or if x = 1 or if x = -1.
That is, 1^1 = 1 and -1^-1 = -1 so in absolute value both equal 1.
Insufficient.

(2) If we're not sure where to start we'll try easy numbers. If x = 1 then 1^5 = 1^1 = 1 so x = 1 works. If x = 2 then 2^5 = 32 but 2^2 = 4 so x = 2 doesn't work. Since the exponent on the left is 5 it makes sense to try x = 5 which gives 5^5 = 5^5. So x = 1 and x = 5 both work.
Insufficient.

Combined:
x = 1 fits both criteria as shown above. How about x = -1? -1^5 = -1 and -1^-1 = -1. So both x = 1 and x = -1 fit both criteria.
Insufficient.

(E) is our answer.

zmtalha · Answer

My answer:E

St1: X=1 or x=-1
St2: x=1 or x=-1

Sent from my iPhone using  GMAT Club Forum mobile app

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

sahuiitd · Answer

Hi Bunuel Sir.... Can you let me what is the mistake in my method .

vamkrispk · Answer

as per statement 2 .. can we not use the common base concept ?? then x=5 would be the answer and would be sufficient ??

TestPrepUnlimited · Answer

We have to think outside of the box for this one. The base here can be x =1 and any exponent of 1 is still 1. Similarly for -1, as long as the exponents are both even/odd.

TestPrepUnlimited · Answer

Analyzing the question:

We are playing around with exponents of exponents. When evaluating exponents we want to keep a few special cases in mind and test them out: The exponent being 0 or 1, the base being 0 or 1. Here we should also test -1 since we see absolute values.

Statement 1: 
x = 1 or x = -1 satisfy this, insufficient.

Statement 2: 
This one is a little tricky, x = 5 and x = 1 both satisfy this equation. Since 5 is an odd exponent, we should test out x=-1 as well, which works.

Combined: 
x = 1 or x = -1 satisfy both statements, insufficient.

Ans: E

Kinshook · Answer

(1)  |x^x| = 1 
x = 1 or x = -1
NOT SUFFICIENT

(2)  x^5 = x^x  
x = 1 or x = -1 or x=5
NOT SUFFICIENT

(1) + (2)
(1)  |x^x| = 1 
x = 1 or x = -1
(2)  x^5 = x^x  
x = 1 or x = -1 or x=5
x = 1 or x = -1
NOT SUFFICIENT

IMO E

exc4libur · Answer

(1)  |x^x| = 1  :  insufic.

|x^x| = 1…x=1…|1^1|=1 
 |x^x| = 1…x=-1…|-1^{-1}|=1

(2)  x^5 = x^x  :  insufic.

x=1: x^5 = x^x…1^5=1^1…1=1…|1^1|=1 
 x=-1: x^5 = x^x…-1^5=-1^{-1}…-1=-1…|-1^{-1}|=1

(1&2) :  insufic.

x=1: x^5 = x^x…1^5=1^1…1=1…|1^1|=1 
 x=-1: x^5 = x^x…-1^5=-1^{-1}…-1=-1…|-1^{-1}|=1

Answer (E)

suchita2409 · Answer

If x ≠ 0, what is the value of x ?

(1) |x^x|=1
x^x could be +1 or -1
Insufficient

(2) x^5=x^x
x could be 1,-1, or 5.
Insufficient

1)+2)
We still have 1 and -1 both
Insufficient 
E

Bunuel · Answer

Official Solution:

If  x ≠ 0 , what is the value of  x  ? 
 
(1)  |x^x| = 1 
 
If  x^x = 1 , then  x=1 .
 
If  x^x = -1 , then  x=-1 .
 
Not sufficient.
 
(2)  x^5 = x^x 
 
Obviously  x=1  ( 1^5=1^1 ) and  x=5  ( 5^5=5^5 ) work. Since 5 is an odd exponent, let's check whether  x=-1  work:  (-1)^5=(-1)^{(-1)} . It does. So, from this statement  x=1 ,  x=5  or  x=-1 .
 
Not sufficient.
 
(1)+(2)  x  still can be 1 or -1. Not sufficient.

Answer: E

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If x 0, what is the value of x ? (1) |x^x| = 1 (2) x^5 = x^x

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