Help with Inequality with Exponents question

[magic_castle32]

How do you prove the following?

7^{\sqrt{5}} > 5^{\sqrt{7}}

Without calculators or working out {\sqrt{5}}, {\sqrt{7}}, of course.

 

[l'Hôpital]

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

So we have
<br /> 7^5 \ and \ 5^{\sqrt{35}}<br />

Now, consider the following:

<br /> 6 = \sqrt{36} &gt; \sqrt{35}<br />

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