WordGPT

Which is bigger, 3293^{29} or 29329^{3}?

3293^{29} is bigger: 3293^{29} = 6.86303 × 101310^{13} and 29329^{3} = 24389.

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 = 3x = 29peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 3^29 against 29^3 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: 3293^{29} = 68630377364883 and 29329^{3} = 24389.

  2. 2

    So 3293^{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 3 and 29 are past e, where the function is decreasing, so the larger exponent loses and 3293^{29} > 29329^{3}.

    ln⁡33 > ln⁡2929\frac{\ln 3}{3}\ >\ \frac{\ln 29}{29}
LaTeX source of the formulas on this page
\[ 3^{29} = 68630377364883\ >\ 29^{3} = 24389 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger, 3293^{29} or 29329^{3}?

3293^{29} is bigger: 3293^{29} = 6.86303 × 101310^{13} and 29329^{3} = 24389.

Similar comparisons

    Which is bigger, 3^29 or 29^3? 3^29 | WordGPT