WordGPT

Which is bigger, 9189^{18} or 18918^{9}?

9189^{18} is bigger: 9189^{18} = 1.50094 × 101710^{17} and 18918^{9} = 198359290368.

Explain it step by step in chatMake a short video of it
The curve ln x / x, which decides every comparison of a^b with b^a2468101214161820-0.10.10.20.30.4x = 9x = 18peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 9^18 against 18^9 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. 1

    Compute both exactly: 9189^{18} = 150094635296999121 and 18918^{9} = 198359290368.

  2. 2

    So 9189^{18} is bigger.

  3. 3

    The reason: taking logs, aba^{b} against bab^{a} is the same as ln⁡aa\frac{\ln a}{a} against ln⁡bb\frac{\ln b}{b}. The function ln⁡xx\frac{\ln x}{x} rises until x=e≈2.718x = e \approx 2.718 and falls afterwards. Both 9 and 18 are past e, where the function is decreasing, so the larger exponent loses and 9189^{18} > 18918^{9}.

    ln⁡99 > ln⁡1818\frac{\ln 9}{9}\ >\ \frac{\ln 18}{18}
LaTeX source of the formulas on this page
\[ 9^{18} = 1.50094\times 10^{17}\ >\ 18^{9} = 198359290368 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 18}{18} \]

Questions

Which is bigger, 9189^{18} or 18918^{9}?

9189^{18} is bigger: 9189^{18} = 1.50094 × 101710^{17} and 18918^{9} = 198359290368.

Similar comparisons

    Which is bigger, 9^18 or 18^9? 9^18 | WordGPT