WordGPT

Which is bigger, 101110^{11} or 111011^{10}?

101110^{11} is bigger: 101110^{11} = 100000000000 and 111011^{10} = 25937424601.

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^a2468101214-0.4-0.20.20.4x = 10x = 11peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 10^11 against 11^10 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: 101110^{11} = 100000000000 and 111011^{10} = 25937424601.

  2. 2

    So 101110^{11} 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 10 and 11 are past e, where the function is decreasing, so the larger exponent loses and 101110^{11} > 111011^{10}.

    ln⁡1010 > ln⁡1111\frac{\ln 10}{10}\ >\ \frac{\ln 11}{11}
LaTeX source of the formulas on this page
\[ 10^{11} = 100000000000\ >\ 11^{10} = 25937424601 \]
\[ \frac{\ln 10}{10}\ >\ \frac{\ln 11}{11} \]

Questions

Which is bigger, 101110^{11} or 111011^{10}?

101110^{11} is bigger: 101110^{11} = 100000000000 and 111011^{10} = 25937424601.

Similar comparisons

    Which is bigger, 10^11 or 11^10? 10^11 | WordGPT