SOLUTION: how would you solve this proof? {{{ (sqrt(6)-sqrt(2))/4 = sqrt(2-sqrt(3))/2 }}}

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Question 293344: how would you solve this proof?
+%28sqrt%286%29-sqrt%282%29%29%2F4+=+sqrt%282-sqrt%283%29%29%2F2+
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
+%28sqrt%286%29-sqrt%282%29%29%2F4+=++sqrt%282-sqrt%283%29%29%2F2+ Start with the given equation.


+sqrt%286%29-sqrt%282%29+=++2%2Asqrt%282-sqrt%283%29%29+ Multiply both sides by the LCD 4 to clear out the fractions.


+sqrt%286%29-sqrt%282%29+=++sqrt%284%29%2Asqrt%282-sqrt%283%29%29+ Rewrite 2 as sqrt%284%29


+sqrt%286%29-sqrt%282%29+=++sqrt%284%282-sqrt%283%29%29%29+ Combine the roots on the right side using the identity sqrt%28x%29%2Asqrt%28y%29=sqrt%28x%2Ay%29


+sqrt%286%29-sqrt%282%29+=++sqrt%288-4%2Asqrt%283%29%29+ Distribute and multiply


sqrt%288-4%2Asqrt%283%29%29+=+sqrt%286%29-sqrt%282%29+ Rearrange the equation.


We essentially have a nested radical on the left side. In general, we have an equation in the form sqrt%28a%2Bb%2Asqrt%28c%29%29=sqrt%28d%29%2Bsqrt%28e%29. The goal from here is to square both sides, gather like terms, and equate rational and irrational parts. So let's do these next tasks.


sqrt%288-4%2Asqrt%283%29%29+=+sqrt%286%29-sqrt%282%29+ Start with the given equation.


8-4%2Asqrt%283%29+=+%28sqrt%286%29-sqrt%282%29%29%5E2+ Square both sides.


FOIL


8-4%2Asqrt%283%29+=+6-2%2Asqrt%286%29%2Asqrt%282%29%2B%28sqrt%282%29%29%5E2+ Square sqrt%286%29 to get 6.


8-4%2Asqrt%283%29+=+6-2%2Asqrt%286%29%2Asqrt%282%29%2B2+ Square sqrt%282%29 to get 2.


8-4%2Asqrt%283%29+=+8-2%2Asqrt%286%29%2Asqrt%282%29+ Combine like terms.


8-4%2Asqrt%283%29+=+8-2%2Asqrt%286%2A2%29+ Combine the roots on the right side using the identity sqrt%28x%29%2Asqrt%28y%29=sqrt%28x%2Ay%29


8-4%2Asqrt%283%29+=+8-2%2Asqrt%2812%29+ Multiply


8-4%2Asqrt%283%29+=+8-2%2Asqrt%284%2A3%29+ Factor 12 into 4*3 (take note that 4 is a perfect square)


8-4%2Asqrt%283%29+=+8-2%2Asqrt%284%29%2Asqrt%283%29+ Break up the root using the identity given above.


8-4%2Asqrt%283%29+=+8-2%2A2%2Asqrt%283%29+ Take the square root of 4 to get 2.


8-4%2Asqrt%283%29+=+8-4%2Asqrt%283%29+ Multiply


From here, we can see that the two numbers are equal since the rational parts are equal and the irrational parts are equal. So this confirms that +%28sqrt%286%29-sqrt%282%29%29%2F4+=++sqrt%282-sqrt%283%29%29%2F2+


Note: Ideally, you'll only manipulate one side to transform it into the other. However, this is much more difficult to do with nested radicals. I find that algebraic manipulations suffice in this case. As a final check, you can use a calculator to find that both sides approximate to the value 0.258819045 (which helps verify the answer).