WordGPT

Which is bigger, 151615^{16} or 161516^{15}?

151615^{16} is bigger: 151615^{16} = 6.56840 × 101810^{18} and 161516^{15} = 1.15292 × 101810^{18}.

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^a24681012141618-0.2-0.10.10.20.30.4x = 15x = 16peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 15^16 against 16^15 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: 151615^{16} = 6568408355712890625 and 161516^{15} = 1152921504606846976.

  2. 2

    So 151615^{16} 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 15 and 16 are past e, where the function is decreasing, so the larger exponent loses and 151615^{16} > 161516^{15}.

    ln⁡1515 > ln⁡1616\frac{\ln 15}{15}\ >\ \frac{\ln 16}{16}
LaTeX source of the formulas on this page
\[ 15^{16} = 6.56840\times 10^{18}\ >\ 16^{15} = 1.15292\times 10^{18} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 16}{16} \]

Questions

Which is bigger, 151615^{16} or 161516^{15}?

151615^{16} is bigger: 151615^{16} = 6.56840 × 101810^{18} and 161516^{15} = 1.15292 × 101810^{18}.

Similar comparisons

    Which is bigger, 15^16 or 16^15? 15^16 | WordGPT