WordGPT

Which is bigger, 162116^{21} or 211621^{16}?

162116^{21} is bigger: 162116^{21} = 1.93428 × 102510^{25} and 211621^{16} = 1.43056 × 102110^{21}.

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^a246810121416182022240.10.20.30.4x = 16x = 21peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 16^21 against 21^16 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: 162116^{21} = 19342813113834066795298816 and 211621^{16} = 1430568690241985328321.

  2. 2

    So 162116^{21} 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 16 and 21 are past e, where the function is decreasing, so the larger exponent loses and 162116^{21} > 211621^{16}.

    ln⁡1616 > ln⁡2121\frac{\ln 16}{16}\ >\ \frac{\ln 21}{21}
LaTeX source of the formulas on this page
\[ 16^{21} = 1.93428\times 10^{25}\ >\ 21^{16} = 1.43056\times 10^{21} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger, 162116^{21} or 211621^{16}?

162116^{21} is bigger: 162116^{21} = 1.93428 × 102510^{25} and 211621^{16} = 1.43056 × 102110^{21}.

Similar comparisons

    Which is bigger, 16^21 or 21^16? 16^21 | WordGPT