For what value of x does (−6)^x = 6^(9−x)?

Question

For what value of x does  (−6)^x = 6^{(9−x)} ?
 
A. −9/2
B. 0
C. 1
D. 9/2
E. There is no real-number solution

vitaliyGMAT · Answer

Hi

a^b = (-a)^b  only if b is even.

In our case, expression to the right is always positive. In order for the expresion to the left to be positive x should be even, but 9-x when x is even will always be odd. 
On the other hand, 9/2 or -9/2 will bring us square root of a negative number - complex number.

Answer E.

sravankumar0911 · Answer

Answer is E, with options A,B,C the exponent part will never be equal on LHS and RHS....for option D the base on both sides are is equal only in mod, hence no real solution exists for the equation

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jfranciscocuencag · Answer

Hello there math experts!!!

What's wrong with doing it like:

(−6)^x = 6^{(9−x)}  ... Let x be 9/2

(−6)^-(9/2) = 6^{(9−(9/2))}    is it possible to cancel out the minus from -6????

(6)^(9/2) = 6^{((18-9)/2))}

(6)^(9/2) = 6^{(9/2)}

ScottTargetTestPrep · Answer

In order for the equation to be true, the exponents must be equal and they must be an even number such as 2 (notice that (-6)^2 = 6^2) or a fraction in lowest terms in the form of m/n where m is even and n is odd such as ⅔ (notice that (-6)^(⅔) = 6^(⅔)). However, setting the exponents equal, we find that:

x = 9 - x

2x = 9

x = 9/2

Since 9/2 is neither of the two cases mentioned above, there is no real-number solution that will allow -6^x to equal 6^(9-x).

Answer: E

jabhatta2 · Answer

Hi  ScottTargetTestPrep  - I am not sure I agree with the yellow.

The LHS of the equation is  un-defined

You cannot have a "negative" WITHIN the bracket because CUBE ROOT of (-6) will be  un-defined

ScottTargetTestPrep · Answer

Cube roots and other odd-index roots of negative numbers are well defined. For instance, the cube root of -8 is -2, and the cube root of -27 is -3. The fifth root of -32 is -2. The eleventh root of -1 is -1. I think you are confusing this concept with square roots and other even-index roots, both of which are indeed not defined for negative numbers.

jabhatta2 · Answer

Hi  ScottTargetTestPrep  (-8) ^ (1/3) is  certainly defined

However

(-8)^(2/3) is  NOT defined

In excel and calculators  (-8)^(2/3)  shows up as  un-defined

According to your logic -  (-8)^(2/3)  must be (+4)

But excel / google / calculator DOES NOT say so

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For what value of x does (−6)^x = 6^(9−x)?

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