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

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GMAT Panda · Accepted Answer

14 We see unknowns in exponents, so we use "Prime Matching" so we start by expressing each side of the equation in prime factors 12^x = (2^23)^x = 2^2x3^x 4^2x+1 = (2^2)^2x+1 = 2^4x+2 We combine like bases on the left side 12^x4^2x+1 = 2^2x3^x2^4x+2 = 2^6x+23^x We then match the numbers of exponents of each prime, starting with the 3s 2^6x+23^x = 2^k3^2 3-exponent: x = 2 We then match the exponent of 2 and substitute x=2 6x+2 = k 6(2)+2 = k k = 14

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$We see unknowns in exponents, so we use "Prime Matching" so we start by expressing each side of the equation in prime factors$	$1 2^{x} = (2^{2} \times 3)^{x} = 2^{2 x} \times 3^{x} 4^{2 x + 1} = (2^{2})^{2 x + 1} = 2^{4 x + 2}$
$We combine like bases on the left side$	$1 2^{x} \times 4^{2 x + 1} = 2^{2 x} \times 3^{x} \times 2^{4 x + 2} = 2^{6 x + 2} \times 3^{x}$
$We then match the numbers of exponents of each prime, starting with the 3s$	$2^{6 x + 2} \times 3^{x} = 2^{k} \times 3^{2} 3 -exponent: x = 2$
$We then match the exponent of 2 and substitute x = 2$	$6 x + 2 = k 6 (2) + 2 = k k = 14$

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