If $x$ and $k$ are integers and $(8^x)(12^{x+2})=(... | Free GMAT Practice Question | GMAT Panda

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If x and k are integers and (8^{x}) (1 2^{x + 2}) = (2^{k}) (3^{5}), what is the value of k ?

Explanation	Calculations	Help
$We see unknowns in exponents, so we use "Prime Matching" and start by expressing each factor on the left side in prime powers$	$8^{x} = (2^{3})^{x} = 2^{3 x} 1 2^{x + 2} = (2^{2} \times 3)^{x + 2} = 2^{2 x + 4} \times 3^{x + 2}$	Theory & Tactics Method Card ALG7 Click to view full details
$We combine like bases on the left side$	$(8^{x}) (1 2^{x + 2}) = 2^{3 x} \times 2^{2 x + 4} \times 3^{x + 2} = 2^{5 x + 4} \times 3^{x + 2}$
$We match the exponent of 3 on both sides$	$2^{5 x + 4} \times 3^{x + 2} = 2^{k} \times 3^{5} 3 -exponent: x + 2 = 5 x = 3$	Theory & Tactics Method Card ALG7 Click to view full details
$We match the exponent of 2 and substitute x = 3$	$5 x + 4 = k 5 (3) + 4 = k k = 19$	Theory & Tactics Method Card ALG7 Click to view full details

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C

If x and k are integers and (8^{x}) (1 2^{x + 2}) = (2^{k}) (3^{5}), what is the value of k ?

Explanation	Calculations	Help
$We see unknowns in exponents, so we use "Prime Matching" and start by expressing each factor on the left side in prime powers$	$8^{x} = (2^{3})^{x} = 2^{3 x} 1 2^{x + 2} = (2^{2} \times 3)^{x + 2} = 2^{2 x + 4} \times 3^{x + 2}$	Theory & Tactics Method Card ALG7 Click to view full details
$We combine like bases on the left side$	$(8^{x}) (1 2^{x + 2}) = 2^{3 x} \times 2^{2 x + 4} \times 3^{x + 2} = 2^{5 x + 4} \times 3^{x + 2}$
$We match the exponent of 3 on both sides$	$2^{5 x + 4} \times 3^{x + 2} = 2^{k} \times 3^{5} 3 -exponent: x + 2 = 5 x = 3$	Theory & Tactics Method Card ALG7 Click to view full details
$We match the exponent of 2 and substitute x = 3$	$5 x + 4 = k 5 (3) + 4 = k k = 19$	Theory & Tactics Method Card ALG7 Click to view full details

Scroll horizontally to view all columns

C