which-number-is-greater-22-55-and-55-22-

Question Number 172323 by Giantyusuf last updated on 25/Jun/22

w h i c h n u m b e r i s g r e a t e r

22^{55} a n d 55^{22} ? ? ?

Answered by nurtani last updated on 25/Jun/22

22^{55} > 55^{22}

Answered by floor(10²Eta[1]) last updated on 25/Jun/22

2^{55} . 11^{55} > 5^{22} . 11^{22}

2^{55} . 11^{33} > 5^{22}

clearly true since 11^{33} > 5^{33} > 5^{22}

\Rightarrow 22^{55} > 55^{22}

Answered by Gazella thomsonii last updated on 25/Jun/22

A = 22^{55} B = 55^{22}

lnA = 55 \ln (22) \ln (B) = 22 \ln (55)

\ln (22) \approx 3.091 \ln (55) \approx 4.007

lnA = 165 . xxx lnB = 88 . xxx

∴ B < A

Answered by MJS_new last updated on 25/Jun/22

f (x) = x^{\frac{1}{x}} has its maximum at x = e \Rightarrow

\forall a, b ∣ e < a < b : a^{\frac{1}{a}} > b^{\frac{1}{b}}

\Leftrightarrow

\frac{\ln a}{a} > \frac{\ln b}{b}

\Leftrightarrow

b \ln a > a \ln b

\Leftrightarrow

a^{b} > b^{a}

for a, b \in N \land a, b > 1 the only exceptions are

2^{3} < 3^{2}

2^{4} = 4^{2}

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