WordGPT

Which is bigger, 9299^{29} or 29929^{9}?

9299^{29} is bigger: 9299^{29} = 4.71012 × 102710^{27} and 29929^{9} = 1.45071 × 101310^{13}.

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^a510152025300.10.150.20.250.30.350.4x = 9x = 29peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 9^29 against 29^9 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: 9299^{29} = 4710128697246244834921603689 and 29929^{9} = 14507145975869.

  2. 2

    So 9299^{29} 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 29 are past e, where the function is decreasing, so the larger exponent loses and 9299^{29} > 29929^{9}.

    ln⁡99 > ln⁡2929\frac{\ln 9}{9}\ >\ \frac{\ln 29}{29}
LaTeX source of the formulas on this page
\[ 9^{29} = 4.71012\times 10^{27}\ >\ 29^{9} = 14507145975869 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger, 9299^{29} or 29929^{9}?

9299^{29} is bigger: 9299^{29} = 4.71012 × 102710^{27} and 29929^{9} = 1.45071 × 101310^{13}.

Similar comparisons

    Which is bigger, 9^29 or 29^9? 9^29 | WordGPT