Next step in solving an equations...

Question

I have managed so far to break down an the following equation:

$x^n+y^n=1$

to

$x^n=1-y^n$

but what is the next step to get $x$ on it's own?

I have hopped over here from StackOverflow where I am trying draw superellipse where a and b are always 1. So applogies for my lack of terminology! I have a very beginner understanding of mathematics and almost no understanding of mathematical notation for equations!

Hint: try the n-th root of each side? Try it with $n = 2$ and then generalize. — Amzoti, Jul 12 '13 at 21:15

amWhy · Accepted Answer · 2013-07-12T21:25:40.730

7

$$x^n+y^n=1$$ $$\iff x^n = 1 - y^n$$

$$\implies \large (x^n)^{\frac 1n} = x = \left(1 - y^n\right)^{\frac 1n}$$

Another way of expressing "to the $1/n$th power" is "the nth root of", which is denoted on the right-hand side below:

$$x = \left(1 - y^n\right)^{\large \frac 1n} = \sqrt[\Large n]{1 - y^n}$$

edited Jul 12 '13 at 21:25

answered Jul 12 '13 at 21:18

amWhy

209,954

Factastic thank you!. The ⟺ symbol is new to me. What is it called so I may google it? – Ross Jul 12 '13 at 21:25
@Ross http://en.wikipedia.org/wiki/If_and_only_if – Ruslan Jul 12 '13 at 21:27
It's used to denote, e.g., "if and only if": it is usually called the "biconditional". $P \iff Q$ is equivalent to $[(P\implies Q);\text{and}; (Q \implies P)]$ – amWhy Jul 12 '13 at 21:27
@amWhy: Looks spot on to me +1 – Amzoti Jul 13 '13 at 01:01

Next step in solving an equations...

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