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The cube root of what integer power of 2 closest to 50?
A. 16
B. 17
C. 18
D. 19
E. 20
OA B
The cube root of what integer power of 2 closest to 50?
A. 16
B. 17
C. 18
D. 19
E. 20
OA B
Say the integer power of 2 closest to 50 is n; thus, we have \(\sqrt[3]{^{2^n}}= 50\)
\({2^{n/3} = 50}\)
We know that 2^5 = 32 < 50 and 2^6 = 64 > 50
Thus, 5 < n/3 < 6 => 15 < n < 18. Thus, n could be 16 or 17. Still we have two options A and B.
• Taking n = 16:
\({2^{n/3} = 50 => 2^{16/3} = 2^5*\sqrt[3]{2} = 32*\sqrt[3]{2}}\)
• Taking n = 17:
\({2^{n/3} = 50 => 2^{17/3} = (2^6)/(\sqrt[3]{2}) = 64/\sqrt[3]{2}}\)
Now we have to compare \(32*\sqrt[3]{2}\) and \(64/\sqrt[3]{2}\). Since 32 is 18 away from 50 and 64 is 14 away from 50, the value of \(64/\sqrt[3]{2}\) must be closer to 50 than \(32*\sqrt[3]{2}\) is closer to 50. Thus, n = 17.
The correct answer: B
Hope this helps!
-Jay
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