Powers & inequalities

Algebra Level 2

Which of the following inequalities is true ?

6 50 < 2 50 ⋅ 3 51 6^{50}< 2^{50} \cdot 3^{51} 4 34 − 1 > 2 69 4^{34}-1>2^{69} ( 2 3 ) 40 < 2 35 ⋅ 3 − 41 \big(\frac{2}{3}\big)^{40}<2^{35}\cdot 3^{-41} 5 304 < 2 304 + 3 300 + 2 6 ⋅ 3 7 5^{304}< 2^{304} + 3^{300} + 2^6\cdot3^7 5 5 − 36 > ( 5 4 20 ) − 1 55^{-36}>(54^{20})^{-1}

Vittorio Bertoletti by Vittorio Bertoletti , 26, Italy

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2 solutions

Aryan Sanghi
Mar 11, 2018

3 51 ^{51} =3 50 ^{50} ×3

2 50 ^{50} ×3 51 ^{51} =2 50 ^{50} ×3 50 ^{50} ×3

=6 50 ^{50} ×3

Hence 6 50 ^{50} <2 50 ^{50} ×3 51 ^{51}

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6 50 = ( 2 ⋅ 3 ) 50 = 2 50 ⋅ 3 50 < 2 50 ⋅ 3 51 6^{50}=(2\cdot 3)^{50}=2^{50}\cdot3^{50}<2^{50}\cdot 3^{51}

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Both of our solutions posted at the same time, what a coincidence

Aryan Sanghi - 3 years, 3 months ago

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