If 8^0.5y 3^0.75x = 12^n then what is the value of x?? 1) n

Question

If 8^(0.5y) * 3^(0.75x) = 12^n then what is the value of x??

(1) n = 3.
(2) Both x and y are natural numbers.

Bunuel · Answer

Not a GMAT question.

8^{\frac{y}{2}}*3^{\frac{3x}{4}}=12^n  -->  2^{\frac{3y}{2}}*3^{\frac{3x}{4}}=2^{2n}*3^n  -->  2^{\frac{3y}{2}-2n}*3^{\frac{3x}{4}-n}=1 . Now, either powers of 2 and 3 are both zero or these two multiples are reciprocals of each other and in this case powers of 2 and/or 3 must be some irrational numbers.

(1) n = 3.
(2) Both x and y are natural numbers.

Each statement alone is not sufficient, as we cannot decide which case we have.

(1)+(2) As all variables are integers then the powers of 2 and 3 cannot be irrational numbers thus they must equal to zero:  \frac{3x}{4}-n=\frac{3x}{4}-3=0  -->  x=4 . Sufficient.

Answer: C.

LM · Answer

(2^(3*y/2))*(3^(3*x/4))=(2^6)*(3^3)

3*y/2=6 
 y=4 
 3*x/4=3 
 x=4

Therefore "A"

Buenel, what's the mistake in my approach?

Bunuel · Answer

(1)  n=3  -->  2^{\frac{3y}{2}}*3^{\frac{3x}{4}}=2^6*3^3  --> now you could equate the powers if you knew that both  x  and  y  are integers, but we don't know that.

For example if  y=0  -->  3^{\frac{3x}{4}}=2^6*3^3  and in this case  x  will be some irrational number.

jullysabat · Answer

Hello Bunnel,

You have mentioned this is not a GMAT Question..
May I know the reason.
Because I find this pattern similar to a GMAT Question.

vjsharma25 · Answer

Bunuel, I couldn't understand your statement "you could equate the powers if you knew that both X and Y are integers, but we don't know that ".
 RHS and LHS are both represented in their respective prime factors. So what's the concept behind not equating the powers in this case?

Bunuel · Answer

(1) n=3 --> 2^{\frac{3y}{2}}*3^{\frac{3x}{4}}=2^6*3^3 --> now you could equate the powers if you knew that both x and y are integers, but we don't know that. For example if y=0 --> 3^{\frac{3x}{4}}=2^6*3^3 and in this case x will be some irrational number. Basically 2^{\frac{3y}{2}}*3^{\frac{3x}{4}}=2^6*3^3 has infinitely many solutions for x and y : for any y there will exist some x to satisfy this equation. If we were told that both are integers then these equation would have only one integer solution: x=4 and y=4 Similar question: disagree-with-oa-106047.html

vjsharma25 · Answer

So important takeaway from this question is you can not equate the powers in an equality if you do not know whether the variables making the powers are integers or not.
Can we say that?

brs1cob · Answer

question is not properly presented and i got it wrong as i assumed some other expression

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If 8^0.5y 3^0.75x = 12^n then what is the value of x?? 1) n

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