Rationalization of 262+3+5My approach:I tried to rationalize the denominator by multiplying it by 2−3−52−3−5. And got the result to be (after a long calculation):24+40−1612+5which is totally not in accordance with the answer, 2+3−5Can someone please explain this/give hints to me.

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Rationalization of 262+3+5My approach:I tried to rationalize the denominator by multiplying it by  2−3−52−3−5. And got the result to be (after a long calculation):24+40−1612+5which is totally not in accordance with the answer, 2+3−5Can someone please explain this/give hints to me.

haiguetenteme7zyu · Accepted Answer

The first term multiplied by the conjugate of the last two terms is what I would do. The following expression has been highlighted in order to make it easier to understand.262+3+5⋅2+3−52+3−526(2+3−5)(2+3+5)(2+3−5)You may be wondering why I do this. Note the formula for the difference of squares:a2−b2=(a+b)(a−b)I am actually letting&amp;nbsp;a=2+3&amp;nbsp;and&amp;nbsp;b=5. Therefore, our fraction can be rewritten as:262+263−265(2+3)2−(5)2=43+62−2302+26+3−5Oh. How nice. The integers in the denominator cancel out!43+62−23026Multiply by&amp;nbsp;262643+62−23026⋅2626=836+1226−4306(26)2=242+243−24524=24(2+3−5)24Cancel 24 out in the numerator and denominator and you get:2+3−5∴d262+3+5=2+3−5In reality, there is a far shorter route. Returning to the fraction262+263−265(2+3)2−(5)2=262+263−2652+223+3−5=262+263−26526Do you see that we can factor out&amp;nbsp;26&amp;nbsp;in the numerator and the denominator out, and you get:2+3−5∴262+3+5=2+3−5

Avery Maxwell · Accepted Answer

Your &quot;long computation&quot; was obviously incorrect because the correct denominator is(2+3+5)(2−(3+5))==22−(3+5)2=2−(3+5+215)=−6−215

Rationalization of \(\displaystyle{\frac{{{2}\sqrt{{{6}}}}}{{\sqrt{{{2}}}+\sqrt{{{3}}}+\sqrt{{{5}}}}}}\) My approach: I tried to rationalize

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