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I picked up a copy of Principles of Mathematical Analysis by Rudin. It just so happened to be the second edition which is drastically different from the third and I can't find solutions to a lot of exercises.

I'd like some hints on how to complete the following few problems.

  1. If $x>0$, $y>0$, and $n$ is a positive integer, prove that $\sqrt[n]{x}\sqrt[n]{y} = \sqrt[n]{xy}$.

I can't seem to figure out where to start with this one. I assume I prove the inequality each way.

  1. If $x>0$, and r is rational ($r = n/m$), define $x^r = \sqrt[m]{x^n}$. Prove that $x^r = \sqrt[m]{x}^n$.

  2. If $x>1$, prove that $x^p < x^q$ whenever $p < q$, $p, q$ are rational.

For this one I wrote p, q as fractions and used 2.

stochasticats
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1 Answers1

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  1. I presume $\sqrt[n]x$ is defined as the unique solution of $y^n=x$ with $y>0$ (and I presume that Rudin has proved there is a unique solution). Then to prove $\sqrt[n]{xy}=\sqrt[n]x\sqrt[n]y$ all you need is to verify that $\sqrt[n]x\sqrt[n]y>0$ and $(\sqrt[n]x\sqrt[n]y)^n=xy$.

  2. Again you need that $(\sqrt[m]x)^n>0$ and $((\sqrt[m]x)^n)^m=x^n$.

  3. I would put $p$ and $q$ as fractions over a common denominator and try to reduce to the case where they are integers.

Angina Seng
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