Question

Show that [(Sqrt(100)]^[(Sqrt(101)] is larger than [(Sqrt(101)]^[(Sqrt(100)] without evaluating the numbers.

Answer 1

We want to show that

Taking the -th power of both sides, we want to show that

Looking at the forms of these numbers, consider the general function

And what we want to show is that

If we can show that is monotonically decreasing, it will be sufficient as then it follows that

We have

So that

That is,

So we get

Thus, we have as

So we have meaning that the function is decreasing for

Thus, in particular for we must have

This means so that as required

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