Find the value of [m][fraction]1/([square_root]1[/square_root])+[squar

Any time you see square roots in a denominator, you must "rationalize the denominator", which means you must get the roots out of the denominator, by multiplying the numerator and denominator by identical (well-chosen) things. When the denominator contains a sum or subtraction, we need to use the difference of squares pattern to eliminate the root or roots. Looking at the first fraction, if we multiply the top and bottom by √2 - √1, we'll get rid of the root in the denominator (and in fact the denominator will become 1).

edit: not sure the math code is working so I'll type without it:

[ 1/(√2 + √1) ] * [ (√2 - √1) / (√2 - √1) ] = (√2 - √1)/ (2 - 1) = √2 - √1


Similarly, the second fraction will equal √3 - √2, the third will equal √4 - √3, and so on, until we reach √100 - √99. So the sum will end up looking like, if we add from right to left:

√100 - √99 + √99 - √98 + √98 - √97 + ... + √3 - √2 + √2 - √1

and everything vanishes except the first and last terms, so the sum equals √100 - √1 = 10 - 1 = 9.

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