Today I was going to post a problem but i suddenly got confused about problem.The problem was if \(\sqrt { \sqrt { 14-\sqrt { 132 } } +\sqrt { 23-\sqrt { 528 } + } \sqrt { 3 } } =a\).Find the value of \(a\).

My solution is following:

\( \sqrt { 14-\sqrt { 132 }}=3+11-2\sqrt { 132 }=\sqrt { { \left( \sqrt { 3 } -\sqrt { 11 } \right) }^{ 2 } } \)

Similarly

\(\sqrt { 23-\sqrt { 528 } }=\sqrt { { \left( \sqrt { 11 } -\sqrt { 12 } \right) }^{ 2 } } \)

And \(\sqrt { 3 }=\sqrt {15-12}=\sqrt { 12+3-6\times 2}=\sqrt { \sqrt { { 12 }^{ 2 } } +\sqrt { { 3 }^{ 2 } } -2\sqrt { 36 }} =\sqrt { { \left( \sqrt { 12 } -\sqrt { 3 } \right) }^{ 2 } } \)

Therefore

\(\sqrt { \sqrt { 14-\sqrt { 132 } } +\sqrt { 23-\sqrt { 528 } + } \sqrt { 3 } } =\sqrt { { \left( \sqrt { 3 } -\sqrt { 11 } \right) }^{ 2 } } +\sqrt { { \left( \sqrt { 11 } -\sqrt { 12 } \right) }^{ 2 } } +\sqrt { { \left( \sqrt { 12 } -\sqrt { 3 } \right) }^{ 2 } } =0\)

But now use calculator and evaluate i get answer above 0.

Please help me about this.Which step did I do wrong.

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## Comments

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TopNewestConvention is that \(\sqrt { x } \) returns a positive value, even though it can be both positive and negative. Hence, if we rewrite your expression in this way

\(\sqrt { { (\sqrt { 11 } -\sqrt { 3 } ) }^{ 2 } } +\sqrt { { (\sqrt { 12 } -\sqrt { 11 } ) }^{ 2 } } +\sqrt { { (\sqrt { 12 } -\sqrt { 3 } ) }^{ 2 } } \)

you'll get a non-zero value which agrees with the original expression.

This is one of the big problems with the idea that a "function can only return one value". By insisting that functions can "only return one value", we're missing out the big picture. It's an artificial restriction. I prefer using implicit equations for exactly that reason, it takes care of all the problems with signs. For example, \(y=\sqrt { 1-{ x }^{ 2 } } \) is not even a circle, it's only a half of a circle. But \({ y }^{ 2 }=1-{ x }^{ 2 }\) is a circle.

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How did you get above expression?

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The big mistake was \(\sqrt{{x}^{2}}=\left| x \right|\) and not \(x\). Hope this clarifies things a bit.

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It's exactly the same as yours, except that I reversed the order of some of the terms, which I can do since, after being squared, it's the same. That is, for example, in the 2nd line, you could have reversed the terms of the expression on the far right with equal mathematical validity.

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