I am doing a pretty hard problem: $$\frac{\sqrt{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}-\sqrt{\sqrt[4]{27}-\sqrt{\sqrt{3}-1}}}{\sqrt{\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}}}$$ So it is a pretty long and complicated problem. I got stuck though. My idea was to turn $\sqrt[4]{27} =\sqrt[4]{3}\sqrt{3}$ and since I cant make the second part easier with Langranges formula (it doesn't apply to this) I made it $\sqrt[4]{3}-1$. I seemed happy that I was getting somewhere and I thought that I had it but later on I just got stuck primarily by the 1's that I don't know what to do with.
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Have you tried multiplying by the conjugate? – ultrainstinct Oct 31 '17 at 17:36
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@Huffman_Coding In which part? I never seemed to see an opportunity where it would help – RiktasMath Oct 31 '17 at 17:42
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what do you mean with solve? – Dr. Sonnhard Graubner Oct 31 '17 at 17:42
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Just to simplify as much as possible. The answer is $\sqrt{2}$ – RiktasMath Oct 31 '17 at 17:44
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But the problem says: Prove that the number is irrational. – RiktasMath Oct 31 '17 at 17:45
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Not to be pedantic, but you are using the word "solve" incorrectly in the title of your question. You can "solve" an equation (for a variable), or, more loosely, you can "solve" a problem, but you cannot "solve" an expression that represents a number. What is it exactly you want to do with that number? I guess you mean something like "simplify" maybe. – Leonard Blackburn Oct 31 '17 at 17:46
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Edited. I meant simplify – RiktasMath Oct 31 '17 at 17:48
2 Answers
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Let $$x=\frac{\sqrt{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}-\sqrt{\sqrt[4]{27}-\sqrt{\sqrt{3}-1}}}{\sqrt{\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}}}$$
The square of numerator is $$2\sqrt[4]{27} -2\sqrt{\sqrt{27}-(\sqrt{3}-1)}=2\sqrt[4]{27}-2\sqrt{2\sqrt{3}+1}$$
The square of denominator is $\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}$.
Hence $x^2 = 2$. Also note that $x>0$.
pisco
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Wow... After doing all sorts of multiplying and division just to find out the answer is this simple... Thanks! – RiktasMath Oct 31 '17 at 18:05
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Dang!........... (Well, I deleted my answer... this just ... blew my answer out of the water.) – fleablood Oct 31 '17 at 18:16
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\begin{align} &=\frac{\bigg(\sqrt{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}-\sqrt{\sqrt[4]{27}-\sqrt{\sqrt{3}-1}}\bigg)\bigg(\sqrt{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}+\sqrt{\sqrt[4]{27}-\sqrt{\sqrt{3}-1}}\big)}{\sqrt{\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}}\bigg(\sqrt{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}+\sqrt{\sqrt[4]{27}-\sqrt{\sqrt{3}-1}}\big)}\nonumber \\ &=\frac{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}-\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}{\sqrt{\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}}\bigg(\sqrt{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}+\sqrt{\sqrt[4]{27}-\sqrt{\sqrt{3}-1}}\big)}\nonumber\\ &=\frac{2\sqrt{\sqrt{3}-1}}{\sqrt{\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}}\bigg(\sqrt{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}+\sqrt{\sqrt[4]{27}-\sqrt{\sqrt{3}-1}}\big)}\nonumber \end{align} Can you take it from here?
ultrainstinct
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It's times like this when you really appreciate MATHEMATICA or Maple...........lol – Mathemagician1234 Oct 31 '17 at 17:52