WordGPT

Which is bigger, 3303^{30} or 30330^{3}?

3303^{30} is bigger: 3303^{30} = 2.05891 × 101410^{14} and 30330^{3} = 27000.

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 = 30peak at eln x / x
The curve ln x / x, which decides every comparison of a^b with b^a. 3^30 against 30^3 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: 3303^{30} = 205891132094649 and 30330^{3} = 27000.

  2. 2

    So 3303^{30} 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 30 are past e, where the function is decreasing, so the larger exponent loses and 3303^{30} > 30330^{3}.

    ln⁡33 > ln⁡3030\frac{\ln 3}{3}\ >\ \frac{\ln 30}{30}
LaTeX source of the formulas on this page
\[ 3^{30} = 2.05891\times 10^{14}\ >\ 30^{3} = 27000 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger, 3303^{30} or 30330^{3}?

3303^{30} is bigger: 3303^{30} = 2.05891 × 101410^{14} and 30330^{3} = 27000.

Similar comparisons

    Which is bigger, 3^30 or 30^3? 3^30 | WordGPT