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I encountered a question that puzzles me. The task is to find the simplified form of $$x\sqrt[2]{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x\sqrt[6]{x\sqrt[7] {x}}}}}}$$ my answer is $$x^\frac{433}{252}$$ but as I graph the unsimplified nested radical to desmos grapher, the graph covers all real numbers as its domain and all real numbers as its range. But as I graph $$x^\frac{433}{252}$$ the graph only assumes all positive numbers as its domain and $$[0,+infinity)$$ as its range... I check the given and I notice something. If I give a negative value for x,, then evaluates the unsimplified nested radical from the inside, the sign follows the parity of the index... it means that it was smoothly evaluated through all indexes.. resulting to a negative final result. But if I give the same negative value of x to the simplified form that is $$x^\frac{433}{252}$$ the even power will make the given number positive thus the final result will be positive. It contradics theprevious result. I believe I'm missing a fundamental concept here. Any idea would be a great help. Tnx!!

rosa
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2 Answers2

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Suppose $x<0$; then

  1. $\sqrt[7]{x}<0$
  2. $x\sqrt[7]{x}>0$
  3. $\sqrt[6]{x\sqrt[7]{x}}>0$
  4. $x\sqrt[6]{x\sqrt[7]{x}}<0$
  5. $\sqrt[5]{x\sqrt[6]{x\sqrt[7]{x}}}<0$
  6. $x\sqrt[5]{x\sqrt[6]{x\sqrt[7]{x}}}>0$
  7. $\sqrt[4]{x\sqrt[5]{x\sqrt[6]{x\sqrt[7]{x}}}}>0$
  8. $x\sqrt[4]{x\sqrt[5]{x\sqrt[6]{x\sqrt[7]{x}}}}<0$
  9. $\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x\sqrt[6]{x\sqrt[7]{x}}}}}<0$
  10. $x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x\sqrt[6]{x\sqrt[7]{x}}}}}>0$
  11. $\sqrt{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x\sqrt[6]{x\sqrt[7]{x}}}}}}>0$
  12. $x\sqrt{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x\sqrt[6]{x\sqrt[7]{x}}}}}}<0$

So you can define the expression for any $x$, and the result is the same as $$ |x|^{433/252}\operatorname{sgn}x $$

egreg
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The identity $\sqrt[n]{x} = x^{1/n}$ is only valid for nonnegative $x$, exactly because otherwise you run into problems like the one you have here.

In fact, even definining $x^a$ with $a\notin \Bbb Z$ is tricky for negative bases, so comparing it to some root of $x$, which may or may not exist, remains a dubious prospect.

Arthur
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  • oh?? I see... :) so does it mean that the domain of the unsimplified nested radical should only be restricted to positive numbers and zero? But why does the desmos grapher also assumes the negative one? sorry A little bit confuse... – rosa Aug 02 '17 at 13:11
  • The unsimplified radical sould easily be defined for all of $\Bbb R$, because as you point out, if we insert a negative value and nest it up from the inside out, then every time we need to take an even root, we have a positive number there, so there is no problem. However, for negative $x$ there is no good simplification. Although you could "cheat" and go with $|x|^{433/252}\cdot \operatorname{sgn}(x)$, where $\operatorname{sgn}$ is the sign function. – Arthur Aug 02 '17 at 13:19
  • Tnx a lot sir :) I forgot the idea of signum function.. :) – rosa Aug 03 '17 at 10:44