Help with Inequality with Exponents question

magic_castle32

How do you prove the following?

[tex]7^{\sqrt{5}} > 5^{\sqrt{7}}[/tex]

Without calculators or working out [tex]{\sqrt{5}}[/tex], [tex]{\sqrt{7}}[/tex], of course.

l'Hôpital

Well, consider both numbers and raise them both to the power of square root of 5.

So we have
[tex]
7^5 \ and \ 5^{\sqrt{35}}
[/tex]

Now, consider the following:

[tex]
6 = \sqrt{36} > \sqrt{35}
[/tex]

So we can just compare 7^5 and 5^6. I'm sure you can calculate those by hand.