Which is bigger, $21^{22}$ or $22^{21}$ ?

Q: Which is bigger, 21^22 or 22^21?

21^22 is bigger: 21^22 = 1.22694 × 10^29 and 22^21 = 1.55194 × 10^28.

$21^{22}$ is bigger: $21^{22}$ = 1.22694 × $10^{29}$ and $22^{21}$ = 1.55194 × $10^{28}$ .

21^{22} = 1.22694\times 10^{29}\ >\ 22^{21} = 1.55194\times 10^{28}

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The curve ln x / x, which decides every comparison of a^b with b^a. 21^22 against 22^21 is the same question as which of the two marked points is higher; the curve peaks at x = e.

How to work it out

1
Compute both exactly: $21^{22}$ = 122694327386105632949003612841 and $22^{21}$ = 15519448971100888972574851072.
2
So $21^{22}$ is bigger.
3
The reason: taking logs, $a^{b}$ against $b^{a}$ is the same as $\frac{\ln a}{a}$ against $\frac{\ln b}{b}$ . The function $\frac{\ln x}{x}$ rises until $x = e \approx 2.718$ and falls afterwards. Both 21 and 22 are past e, where the function is decreasing, so the larger exponent loses and $21^{22}$ > $22^{21}$ .
$\frac{\ln 21}{21}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 21^{22} = 1.22694\times 10^{29}\ >\ 22^{21} = 1.55194\times 10^{28} \]
\[ \frac{\ln 21}{21}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

21^{22}

22^{21}

$21^{22}$ is bigger: $21^{22}$ = 1.22694 × $10^{29}$ and $22^{21}$ = 1.55194 × $10^{28}$ .

Which is bigger, 212221^{22}2122 or 222122^{21}2221?

How to work it out

Questions

Similar comparisons

Which is bigger, $21^{22}$ or $22^{21}$ ?