If x and k are Integers and \((12^x)(4^{2x+1})=(2)^k(3^2)\), What GMAT Problem Solving

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

1. 5
2. 7
3. 10
4. 12
5. 14

Answer:
Approach Solution (1):
The given expression can be written as:
\((12^x)(4^{2x+1})=(2)^k(3^2)\);
\((2^{2x}*3^x)(2^{2(2x+1}))=(2^k)(3^2);\)
\((2^{2x+2(2x+1)}*3^x)=(2^k)(3^2);\)
Equate the powers of 3, we will get:
x = 2;
Equate the powers of 2, we will get:
2x + 2(2x + 1) = k
6x + 2 = k
k = 14
Correct option: E

Approach Solution (2):
Re-write the given expression as:
\((12^x)(4^{2x+1})=(2)^k(3^2)\)
\(\Rightarrow3^x*4^x*4^{(2x+1)}=(2^k)(3^2)\)
LHS and RHS has only 1 base of 3
So, x = 2
\(4^2.4^5=2^k\)
\(4^{(2+5)} = 2^k\)
\(4^7 = 2^k\)
\(2^{2*7} = 2^k\)
\(\therefore k = 2*7 = 14\)
Correct option: E

Approach Solution (3):
Let’s simplify the given equation:
Equating the exponents with the like bases, we see that:
\((3*2^2)^x*(2^2)^{2x+1}=2^k*3^2\)
\(3^x*2^{2x}*2^{4x+2}=2^k*3^2\)
\(3^x*2^{(6x+2)}=2^k*3^2\)
x = 2 and 6x + 2 = k
Thus, k = 6 (2) + 2 = 14
Correct option: E

“If x and k are integers and\((12^x)(4^{2x+1})=(2)^k(3^2)\), what is the value of k?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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