Problem Solving

M7MBA wrote: ↑

Thu Jul 30, 2020 11:27 am

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]

Solution:

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