Hello to all !

I have a little problem !!

Who is bigger tyen who ????????

Show me with the stage of the solution!

Thank you !

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- Sep 27th 2010, 02:00 PMyehoramWho is bigger !!
Hello to all !

I have a little problem !!

Who is bigger tyen who ????????

Show me with the stage of the solution!

Thank you ! - Sep 27th 2010, 03:10 PMpickslides
I bet your calculator knows. Are you allowed to use it?

__Spoiler__: - Sep 27th 2010, 04:18 PMmelese
Let and . The squares of these two numbers are and .

So it's clear that and therefore .

Now we have and .

Write . Since we must have . - Sep 28th 2010, 07:29 AMArchie Meade
There is no need for any squaring......

and

Then use the property of surd conjugates

Since 7-3=6-2, then (surd)(surd conjugate) is equal for both.

Hence, since

A more general way to look at this.....

For any positive difference "k" and any positive increment "w"

Yet another general way to consider it is as follows.....

Generally as the numbers increase with a common difference, the difference between their square roots keeps decreasing,

since , defined for x>4, becomes insignificantly small as x increases without bound,

while x-(x-4) is always 4.

This means as "n" keeps increasing.

You can use calculus to show that

is a decreasing function as x increases.

Therefore the derivative is negative, so the function is a decreasing function.

Therefore

You could also examine the function

which is defined for x positive.