Number Comparison

Algebra Level 3

If A = 1 7 14 A=17^{14} and B = 3 1 11 B=31^{11} , which of the following is true?

A > B A>B A = B A=B A < B A<B

Jim Chale by Jim chale , 20, Greece

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2 solutions

Jim Chale
Jul 3, 2018

A = 1 7 14 > 1 6 14 = 2 56 > 2 55 = 3 2 11 > 3 1 11 = B A=17^{14}>16^{14}=2^{56}>2^{55}=32^{11}>31^{11}=B ⇒ A > B \Rightarrow A>B

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Naren Bhandari
Jul 3, 2018

A B = 1 7 14 3 1 11 = 1 7 3 { ( 31 − 14 ) 11 3 1 11 } = 1 7 3 { ( 1 − 14 31 ) 11 } ⏟ by binomial expansion > 1 \dfrac{A}{B} = \dfrac{17^{14}}{31^{11}} = 17^{3}\left\{\dfrac{(31-14)^{11}}{31^{11}}\right\} = 17^3\underbrace{\left\{\left(1-\dfrac{14}{31}\right)^{11}\right\}}_{\text{by binomial expansion}} > 1 Therefore, A > B A > B .

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