Number Properties Puzzle – Challenge 10

Challenge:

If \(a=2^{6000}, b=3^{4000}\) and \(c=7^{2000}\), which of the following is true?

A- \(a < b < c\)

B- \(c < b < a\)

C- \(a < c < b\)

D- \(c < a < b\)

E- \(b < a < c\)

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\(a=2^{6000}, b=3^{4000}\) and \(c=7^{2000}\)
Find the \(2000^{th}\) rout of each number:
\(\sqrt[2000]{a}= \sqrt[2000]{2^{6000}} = 2^{\frac{6000}{2000}} = 2^3 = 8\)
\(\sqrt[2000]{b} =\sqrt[2000]{3^{4000}} = 3^{\frac{4000}{2000}} = 3^2 = 9\)
\(\sqrt[2000]{c} = \sqrt[2000]{7^{2000}} = 7^{\frac{2000}{2000}} = 7^1 = 7\)
Therefore: \(c < a < b\)

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