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A cloud-storage provider sets each customer's allotted capacity, in gigabytes, equal to (2ᵏ)² × 2², where k is a positive integer. For one enterprise customer this capacity equals 2¹⁶ gigabytes. What is the value of k?
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Correct answer
B
Method 1 (laws of exponents). First apply the power-of-a-power rule: (2ᵏ)² equals 2²ᵏ, because you multiply the inner exponent by the outer one. Next apply the product rule for same-base powers: 2²ᵏ × 2² equals 2⁽²ᵏ⁺²⁾, because you add the exponents when the bases match. Set this equal to the given capacity: 2⁽²ᵏ⁺²⁾ equals 2¹⁶. Since the bases are equal, the exponents must be equal, so 2k + 2 equals 16. Then 2k equals 14, and k equals 7. So the right answer is (B).
Method 2 (test the choices). The expression is (2ᵏ)² × 2². For k = 7 this is 2¹⁴ × 2², which equals 2¹⁶, matching the stem. Trying a smaller value such as k = 4 gives 2⁸ × 2², or 2¹⁰, which is too small, and a larger value such as k = 14 overshoots, so only k = 7 works. So the right answer is (B).
Why the wrong choices are tempting: (A) 4 comes from multiplying the exponents rather than adding them when the two same-base factors are combined. (C) 8 comes from handling the squared term correctly but forgetting to account for the extra factor of 2². (D) 12 comes from adding the outer exponent instead of multiplying, which misapplies the power-of-a-power rule. (E) 14 comes from dropping the outer square completely. Each distractor is the landing point of one specific exponent-rule slip.
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