If x^y = 1 then what is the value of x?

Question

If  x^y = 1 , then what is the value of x?

(1) x < 0
(2) y is even integer

Princ · Answer

OA: E

Given:  x^y = 1

Statement 1:  x < 0

taking  x = -1 , and  y = 2  , It satisfies the  x^y = 1  as  (-1)^2  is equal to  1 .

taking  x = -2 , and  y = 0  , It satisfies the  x^y = 1  as  (-2)^0  is equal to  1 .

Statement  1  alone is insufficient as we are not getting a unique value of  x .

Statement 2:  y  is even integer

taking  x = 1 , and  y = 2  , It satisfies the  x^y = 1  as  (1)^2  is equal to  1 .

taking  x = 100 , and  y = 0  , It satisfies the  x^y = 1  as  (100)^0  is equal to  1 .

Statement  2  alone is insufficient as we are not getting a unique value of  x .

Combining  (1)  and  (2)  will also be insufficient.

ArjunJag1328 · Answer

Most important element to answer this question is to know that any number raise to "0" is 1. (e.g. 1^0 =1 or -1^0=1)

Now let's work this out.

x^y=1, x=?

St 1: x<0 (x is -ve)

Case 1: -1^2 = 1
Case 2: -2^0 = 1

Therefore X could be -1,-2,-3 or any -ve number if the power is 0. Insufficient AD/BCE

St 2: Y is even

(Note: 0 is an even number)

Case 1: x=1, y=2
1^2=1
Case 2: x=2, y=0
2^0=1

Hence B is insufficient. ~~AD/B~~CE

St1 + St2

x^y=1 (x<0, y is even)

Case 1: x=-1 and y=2
=> -1^2=1

Case 2: x=-2 and y=0
=> -2^0=1

From statement 1 and 2, we have proved that x can have multiple values. ~~AD/BC~~E

Therefore the answer is E

Thank you
Arjun

fskilnik · Answer

{x^y} = 1

? = x

Let´s ALGEBRAICALLY BIFURCATE statements (1) and (2) together, to be "shielded" that the right answer is (E) immediately!

\left( {1 + 2} ight)\,\,\,\left\{ \begin{gathered}
 \,{	ext{Take}}\,\,\left( {x,y} ight) = \left( { - 1,0} ight)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = - 1 \hfill \
 \,{	ext{Take}}\,\,\left( {x,y} ight) = \left( { - 2,0} ight)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = - 2 \hfill \ 
\end{gathered} ight.

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

fskilnik · Answer

Hi ArjunJag1328!

You are correct for any real number... except zero.

More explicitly:

{0^0}\,\,{	ext{is}}\,\,{	ext{not}}\,\,{	ext{defined}}

Curiosity: there are some cases in Mathematics that it is CONVENIENT to admit this expression is also equal to 1 (when you are dealing with multi-indices, for example).

But this is just a "local convention" to help you present,  in that context , formulas in a much more pleasant way...

I hope you find this interesting!

Regards,
Fabio.

P.S.: in the GMAT you will NOT be asked to evaluate 0^0 , of course.

BrentGMATPrepNow · Answer

Target question :    What is the value of x?

Given : x^y = 1  
If x^y = 1, then there are 3 possible cases: 
 case i: x = 1, and y = any value (e.g., 1^9 = 1)
case ii: x = -1, and y = an even integer (e.g., (-1)^4 = 1) 
case iii: x = any non-zero value, and y = 0 (e.g., 7^0 = 1)

Statement 1 : x < 0  
Let's TEST some values.
There are several values of x and y that satisfy statement 1. Here are two: 
 Case a : x = -1, and y = 2. Notice that x^y = (-1)^2 = 1. In this case, the answer to the target question is  x = -1 
 Case b : x = -3, and y = 0. Notice that x^y = (-3)^0 = 1. In this case, the answer to the target question is  x = -3 
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : y is even integer 
Let's TEST some values.

PRO-TIP: When testing values, always check to see if you can reuse previous values.     
In this case, we can reuse BOTH cases: 
 Case a : x = -1, and y = 2. Notice that x^y = (-1)^2 = 1. In this case, the answer to the target question is  x = -1 
 Case b : x = -3, and y = 0. Notice that x^y = (-3)^0 = 1. In this case, the answer to the target question is  x = -3 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
IMPORTANT: Notice that I was able to use the   same counter-examples   to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED. 
In other words, 
 Case a : x = -1, and y = 2. Notice that x^y = (-1)^2 = 1. In this case, the answer to the target question is  x = -1 
 Case b : x = -3, and y = 0. Notice that x^y = (-3)^0 = 1. In this case, the answer to the target question is  x = -3 
Since we cannot answer the  target question  with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

bumpbot · Answer

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If x^y = 1 then what is the value of x?

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