What is the value of k?!?

[gmatpup]

If x and k are integers and (12^x)(4^2x+1) = (2^k)(3^2), what is the value of k?

A. 5

B. 7

C. 10

D. 12

E. 14


I know there is a quick way to solve this, can someone please tell me

Answer is: E

[Anurag@Gurome]

If x and k are integers and (12^x)(4^2x+1) = (2^k)(3^2), what is the value of k?

A. 5
B. 7
C. 10
D. 12
E. 14

I know there is a quick way to solve this, can someone please tell me

Answer is: E

(12^x) * [4^(2x+1)] = (2^k) * (3^2)
(2^2x)*(3^x)* (2^4x) * 2^2 = (2^k) * (3^2)
2^(6x+2) * (3^x) = (2^k) * (3^2)
Since bases are the same, so exponents will be equal.
6x + 2 = k
x = 2
k = (6 * 2) + 2 = 14
The correct answer is E.

[neelgandham]

If you ask me, the answer would be 'it is more to do with practice than anything else'. If the same problem is on my math paper(school level). I would have done it this way.

(12^x)*(4^2x+1) = (2^k)*(3^2)
(4^x)*(3^x)*(4^2x+1) = (2^k)*(3^2)
(2^2x)*(3^x)*(4^2x+1) = (2^k)*(3^2)
(2^2x)*(3^x)*(2^2*(2x+1)) = (2^k)*(3^2)
(2^2x)*(3^x)*(2^4x+2) = (2^k)*(3^2)
(2^2x)*(2^4x+2)*(3^x) = (2^k)*(3^2)
(2^(2x+4x+2))*(3^x) = (2^k)*(3^2)
Bases are same, so we can equate the exponents

(3^x) = (3^2), Implies x = 2;
(2^(2x+4x+2))=(2^k), Implies 6x+2 = k; k = (6*2)+2 = 14 (x = 2)

But if the same question is on the GMAT, I would do it the GMAT way:

As you read through the question, your hand(and the pen of course) with the assistance of your brain(again by practice) should give you an equation as shown below by the time you finish reading the question.

2^(6x+2) * (3^x) = (2^k) * (3^2)
x= 2 and 6x+2 = 14 (shouldn't take more than 45 seconds to solve this question in the GMAT exam.

[gmatpup]

If x and k are integers and (12^x)(4^2x+1) = (2^k)(3^2), what is the value of k?

A. 5
B. 7
C. 10
D. 12
E. 14

I know there is a quick way to solve this, can someone please tell me

Answer is: E

(12^x) * [4^(2x+1)] = (2^k) * (3^2)
(2^2x)*(3^x)* (2^4x) * 2^2 = (2^k) * (3^2)
2^(6x+2) * (3^x) = (2^k) * (3^2)
Since bases are the same, so exponents will be equal.
6x + 2 = k
x = 2
k = (6 * 2) + 2 = 14
The correct answer is E.

Where did you get x =2 ?

Thanks again

[shankar.ashwin]

WHen you compare the bases on both sides of the equation, you have 2^(something) and 3^(something)

When 2 equations have same bases, their powers can be equated. SO you have 3^x in the left and 3^2 in the right. So you get x = 2.

Now you can use that to equate powers of 2.

Where did you get x =2 ?

Thanks again

[gmatpup]

oh wow I totally overlooked that!!!

Thanks sooo much everyone for the clarification, it makes this problem super easy now