If \(a=(2^3)(4^3)(5^9)\) and \(b=(4^6)(5^6)(6^9),\) then which of the following values is less than \(\sqrt[3]{ab}?\)
A. \((2^7)(5^5)(6^3)\)
B. \(2(4^3)(5^5)(6^3)\)
C. \((2^{10})(3^3)(5^5)\)
D. \((2^{12})(5^5)(6)\)
E. \((2^6)(5^3)(6^7)\)
[spoiler]OA=D[/spoiler]
Source: Princeton Review
If \(a=(2^3)(4^3)(5^9)\) and \(b=(4^6)(5^6)(6^9),\) then which of the following values is less than \(\sqrt[3]{ab}?\)
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Solution:M7MBA wrote: ↑Thu Jul 30, 2020 11:27 amIf \(a=(2^3)(4^3)(5^9)\) and \(b=(4^6)(5^6)(6^9),\) then which of the following values is less than \(\sqrt[3]{ab}?\)
A. \((2^7)(5^5)(6^3)\)
B. \(2(4^3)(5^5)(6^3)\)
C. \((2^{10})(3^3)(5^5)\)
D. \((2^{12})(5^5)(6)\)
E. \((2^6)(5^3)(6^7)\)
[spoiler]OA=D[/spoiler]
Since ab = (2^3)(4^9)(5^15)(6^9), 3^√(ab) = (2)(4^3)(5^5)(6^3) = (2^7)(5^5)(6^3) = (2^7)(5^5)(2^3)(3^3) = (2^10)(3^3)(5^5), we see that choices A, B and C are exactly equal to 3^√(ab). This leaves us with either choice D or E as the correct answer. Let’s look at choice D first:
(2^12)(5^5)(6) = (2^13)(3)(5^5) = (2^10)(2^3 x 3)(5^5)
Compare this with (2^10)(3^3)(5^5) (i.e., the last expression of 3^√(ab)), we see that while they both have 2^10 and 5^5, 2^3 x 3 = 24 is less than 3^3 = 27. Therefore, (2^10)(2^3 x 3)(5^5) is less than (2^10)(3^3)(5^5).
Answer: D
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