Question

who is greater among the number 39^(40) and 40^(39)

Answer 1

39⁴⁰ is greater among the two mentioned numbers.

-39⁴⁰=4.3912107510207765091e+63

- 40³⁹ = 3.0223145490365729368e+62

- So, it's clear that 39⁴⁰ has a significantly greater value than the 40³⁹.

Answer 2

39^(40) is greater than 40^(39)

• Let us just compare the two numbers.
• The L.C.M of 39 and 40 is 1560. So we just do the 1560th root operation on both of them

• now $\sqrt[1560]{39^{40} } =39^{\frac{1}{39} }$ and $\sqrt[1560]{40^{39} } =40^{\frac{1}{40} }$

• We know that the function f(x) = $x^{\frac{1}{x} }$ is a decreasing function whenever x>=e
• so f(40) < f(39) or f(39) > f(40)

• notice that f(39) = $39^{\frac{1}{39} }$ and f(40) =  $40^{\frac{1}{40} }$

• so $39^{\frac{1}{39} }$ > $40^{\frac{1}{40} }$
• now raising 1560th power on both sides we get 39^(40) > 40^(39)