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Difficulty: 505-555 Level,   Algebra,   Exponents,                              
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Re: If x and k are integers and (12^x)(4^2x+1)= (2^k)(3^2), what [#permalink]
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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


Sol: The given expression can be reduced to (3^x*4^x)4^(2x+1)= 2^k*3^2

So we see that 3^x=3^2 or x=2

Also 4^x*4^(2x+1) can be written as 4^3x+1 =2^k or 2^2(3x+1)= 2^k or k=2*(3x+1) or k=14

Ans is E

600 level is okay.
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Re: If x and k are integers and (12^x)(4^2x+1)= (2^k)(3^2), what [#permalink]
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Its E..

12=3*2^2

4^2x+1=2^2(2x+1)

well will get x=2..

and 2x+4x+2=k

6x+2=k
x=2
k=14
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Re: If x and k are integers and (12^x)(4^2x+1)= (2^k)(3^2), what [#permalink]
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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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


Let’s simplify the given equation:

(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) = 3^2 * 2^k

Equating the exponents with the like bases, we see that:

x = 2 and 6x + 2 = k

Thus, k = 6(2) + 2 = 14.

Answer: E
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Re: If x and k are integers and (12^x)(4^2x+1)= (2^k)(3^2), what [#permalink]
Bunuel wrote:
SOLUTION

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

\((12^x)(4^{2x+ 1})= (2^k)(3^2)\);

\((2^{2x}*3^x)(2^{2(2x+ 1)})= (2^k)(3^2)\);

\((2^{2x+2(2x+ 1)}*3^x)= (2^k)(3^2)\);

Equate the powers of 3 --> \(x=2\);
Equate the powers of 2 --> \(2x+2(2x+ 1)=k\) --> \(6x+2=k\) --> \(k=14\).

Answer: E.



Hi Bunuel

\((2^{2x+2(2x+ 1)}*3^x)= (2^k)(3^2)\); after this

\((2^{6x+2}*3^x)= (2^k)(3^2)\); <--- i get this

how do you Equate the powers of 3 --> \(x=2\); :? where do you see power of 3 ? :? how you get 2 :?

how do you Equate the powers of 2 --> \(2x+2(2x+ 1)=k\) --> \(6x+2=k\) --> \(k=14\). and get 14 :?

i know this rule -->

if \(a^x = a^y\) then \(x = y\)

in other words if the bases match then exponents are equal.

please explain :) \(thank you ^ {1000}\)

:)
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Re: If x and k are integers and (12^x)(4^2x+1)= (2^k)(3^2), what [#permalink]
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dave13 wrote:
Bunuel wrote:
SOLUTION

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

\((12^x)(4^{2x+ 1})= (2^k)(3^2)\);

\((2^{2x}*3^x)(2^{2(2x+ 1)})= (2^k)(3^2)\);

\((2^{2x+2(2x+ 1)}*3^x)= (2^k)(3^2)\);

Equate the powers of 3 --> \(x=2\);
Equate the powers of 2 --> \(2x+2(2x+ 1)=k\) --> \(6x+2=k\) --> \(k=14\).

Answer: E.



Hi Bunuel

\((2^{2x+2(2x+ 1)}*3^x)= (2^k)(3^2)\); after this

\((2^{6x+2}*3^x)= (2^k)(3^2)\); \(x=2\); :? [color=#ff0000] where do you see power of 3 ? :? how you get 2 :?

how do you Equate the powers of 2 --> \(2x+2(2x+ 1)=k\) --> \(6x+2=k\) --> \(k=14\). and get 14 :?

i know this rule -->

if \(a^x = a^y\) then \(x = y\)

in other words if the bases match then exponents are equal.

please explain :) \(thank you ^ {1000}\)

:)


2^(2x+2(2x+ 1))*3^x= (2^k)*(3^2);

Equate the powers of 3 --> \(x=2\);

Next, equate the powers of 2: \(2x+2(2x+ 1)=k\) --> \(6x+2=k\). SINCE x = 2 (form above), then \(k=14\).
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Re: If x and k are integers and (12^x)(4^2x+1)= (2^k)(3^2), what [#permalink]
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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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


Given: \((12^x)(4^{2x+ 1})= (2^k)(3^2)\)

Rewrite \(12\) and \(4\) as follows: \([(2^2)(3^1)]^x[(2^2)^{2x+ 1}]= (2^k)(3^2)\)

Apply exponent laws to get: \((2^{2x})(3^x)(2^{4x+ 2})= (2^k)(3^2)\)

On the left side, combine the powers of \(2\) to get: \((2^{6x+2})(3^x)= (2^k)(3^2)\)

At this point we can see that \(6x+2 = k\) AND \(x = 2\)

Take: \(6x+2 = k\)

Replace \(x\) with \(2\) to get: \(6(2)+2 = k\)

Evaluate: \(14 = k\)

Answer: E

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Re: If x and k are integers and (12^x)(4^2x+1)= (2^k)(3^2), what [#permalink]
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\(12^x4^{(2x+1)}= (2^k)(3^2)\)
\((3*2^2)^x(2^2)^{(2x+1)}= (2^k)(3^2)\)
\(3^x2^{x+2(2x+1)}= (2^k)(3^2)\)

\(x=2\)
\(k=2x+2(2x+1)\)
\(k=2*2+2(2*2+1)\)
\(k=14\)

Answer E
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If x and k are integers and (12^x)(4^2x+1)= (2^k)(3^2), what [#permalink]
Asked: If x and k are integers and \((12^x)(4^{2x+ 1})= (2^k)(3^2)\), what is the value of k ?

\(3^x*2^{2x}*2^{4x+2} = 2^k*3^2\)
\(2^{6x+2}*3^x = 2^k*3^2\)

Equating the powers of 3 & 2 respectively, we get
x = 2
k = 6x+2 = 14

IMO E
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If x and k are integers and (12^x)(4^2x+1)= (2^k)(3^2), what [#permalink]
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