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

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

Explanation	Calculations	Help
$We see unknowns in exponents, so we use "Prime Matching" so we start by expressing each side of the equation in prime factors$	$1 2^{x} = (2^{2} \times 3)^{x} = 2^{2 x} \times 3^{x} 4^{2 x + 1} = (2^{2})^{2 x + 1} = 2^{4 x + 2}$	Theory & Tactics Method Card ALG7 Click to view full details
$We combine like bases on the left side$	$1 2^{x} \times 4^{2 x + 1} = 2^{2 x} \times 3^{x} \times 2^{4 x + 2} = 2^{6 x + 2} \times 3^{x}$
$We then match the numbers of exponents of each prime, starting with the 3s$	$2^{6 x + 2} \times 3^{x} = 2^{k} \times 3^{2} 3 -exponent: x = 2$	Theory & Tactics Method Card ALG7 Click to view full details
$We then match the exponent of 2 and substitute x = 2$	$6 x + 2 = k 6 (2) + 2 = k k = 14$	Theory & Tactics Method Card ALG7 Click to view full details

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

Explanation	Calculations	Help
$We see unknowns in exponents, so we use "Prime Matching" so we start by expressing each side of the equation in prime factors$	$1 2^{x} = (2^{2} \times 3)^{x} = 2^{2 x} \times 3^{x} 4^{2 x + 1} = (2^{2})^{2 x + 1} = 2^{4 x + 2}$	Theory & Tactics Method Card ALG7 Click to view full details
$We combine like bases on the left side$	$1 2^{x} \times 4^{2 x + 1} = 2^{2 x} \times 3^{x} \times 2^{4 x + 2} = 2^{6 x + 2} \times 3^{x}$
$We then match the numbers of exponents of each prime, starting with the 3s$	$2^{6 x + 2} \times 3^{x} = 2^{k} \times 3^{2} 3 -exponent: x = 2$	Theory & Tactics Method Card ALG7 Click to view full details
$We then match the exponent of 2 and substitute x = 2$	$6 x + 2 = k 6 (2) + 2 = k k = 14$	Theory & Tactics Method Card ALG7 Click to view full details

Scroll horizontally to view all columns

14