If $a$ and $k$ are integers and $18^{a}\times 2^{a... | Free GMAT Practice Question | GMAT Panda

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If a and k are integers and 1 8^{a} \times 2^{a + 5} = 2^{k} \times 3^{8}, what is the value of k ?

Explanation	Calculations	Help
$We recognize that the equation has unknowns in exponents, so we apply "Prime Matching" to compare exponents of like bases.$	$1 8^{a} = (2 \times 3^{2})^{a} = 2^{a} \times 3^{2 a} 2^{a + 5} = 2^{a + 5}$	Theory & Tactics Method Card ALG7 Click to view full details
$We then combine the factors with the same base on the left side.$	$1 8^{a} \times 2^{a + 5} = 2^{a} \times 3^{2 a} \times 2^{a + 5} = 2^{2 a + 5} \times 3^{2 a}$
$We equate the exponent of 3 on both sides.$	$2^{2 a + 5} \times 3^{2 a} = 2^{k} \times 3^{8} 2 a = 8$	Theory & Tactics Method Card ALG7 Click to view full details
$We then match the exponent of 2, substituting a = 4 into the expression for k .$	$2 a + 5 = k 2 \times 4 + 5 = k k = 13$	Theory & Tactics Method Card ALG7 Click to view full details

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C

If a and k are integers and 1 8^{a} \times 2^{a + 5} = 2^{k} \times 3^{8}, what is the value of k ?

Explanation	Calculations	Help
$We recognize that the equation has unknowns in exponents, so we apply "Prime Matching" to compare exponents of like bases.$	$1 8^{a} = (2 \times 3^{2})^{a} = 2^{a} \times 3^{2 a} 2^{a + 5} = 2^{a + 5}$	Theory & Tactics Method Card ALG7 Click to view full details
$We then combine the factors with the same base on the left side.$	$1 8^{a} \times 2^{a + 5} = 2^{a} \times 3^{2 a} \times 2^{a + 5} = 2^{2 a + 5} \times 3^{2 a}$
$We equate the exponent of 3 on both sides.$	$2^{2 a + 5} \times 3^{2 a} = 2^{k} \times 3^{8} 2 a = 8$	Theory & Tactics Method Card ALG7 Click to view full details
$We then match the exponent of 2, substituting a = 4 into the expression for k .$	$2 a + 5 = k 2 \times 4 + 5 = k k = 13$	Theory & Tactics Method Card ALG7 Click to view full details

Scroll horizontally to view all columns

C