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?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given equation.
Multiply both sides by the LCD 4 to clear out the fractions.
Rewrite as
Combine the roots on the right side using the identity
Distribute and multiply
Rearrange the equation.
We essentially have a nested radical on the left side. In general, we have an equation in the form . 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.
Start with the given equation.
Square both sides.
FOIL
Square to get 6.
Square to get 2.
Combine like terms.
Combine the roots on the right side using the identity
Multiply
Factor 12 into 4*3 (take note that 4 is a perfect square)
Break up the root using the identity given above.
Take the square root of 4 to get 2.
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
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).
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