<s>[INST] Algebra question grade 9 I am a a student and I am having difficulty with answering this question. I keep getting the answer wrong. Please may I have a step by step solution to this question so that I won't have difficulties with answering these type of questions in the future.
Question: Rearrange the formula below to make n the subject.
$$x = \sqrt{\frac{1+n}{1-n}}.$$
I got rid of the square roots first:
$$x^2(1-n) = 1+n.$$ 
However I don't know what to do next.
Kind Regards. Help would be appreciated
 [/INST] Your first step is correct. So, we have
$$x^2 = \frac{1+n}{1-n}$$
Now, the question is: how do we get rid of this fraction and simplify two $n$'s to one? The natural step is to cross multiply like this:
\begin{align}
{x^2} \cdot \color{#f00}{1} &= \frac{1+n}{1-n} \cdot \color{#f00}1\\
{x^2} \cdot  \color{red}{\left(\frac{1-n}{1-n} \right)} &= \frac{1+n}{1-n} \cdot  \color{red}{\left(\frac{1-n}{1-n}\right)}
\end{align}
The above is how cross multiplication works. Both sides are always being multiplied by $1$. We can rewrite the 1 as anything as long as the 1 simplifies to 1.
Which gives us what you came up with:
$$x^2 \cdot (1-n) = (1+n)$$
Now, it's a matter of just expanding and bringing all of our $n$'s to one side. 
\begin{align}
x^2 \cdot (1-n) &= (1+n) \\
x^2 - nx^2 &= 1 + n \\
x^2 - 1 &= n + nx^2 \\
...
\end{align}
There are two more steps left which I have left for your to complete.
Some times it pays to just expand even if it looks ugly to you and then try to factor things based on whatever you are trying to rearrange for. 
Let me know if you have any further questions :)
EDIT:
So, you have $x^2 - 1$ on one side and $n + nx^2$ on another side. Since you have all of your $n$'s in one place you can try separate the $n$'s from everything else because remember, you want to isolate for $n$.  You can do this by factoring $n$ from $n + nx^2$.  So you will have $n (1 + x^2)$.  
Since you want $n$ by itself, you can divide both sides by $(1 + x^2)$ and come to your answer. 
What you did is: 
\begin{align}
\require{cancel}
x^2 - 1 &= n + nx^2 \\
\frac{\cancel{x^2} - 1}{\cancel{x^2}} &= \frac{n + n\cancel{x^2}} {\cancel{x^2}} \color{red}{Wrong.} \\
\end{align}
Remember, when you have more than one term in the numerator like you do above and divide by $1$ thing i.e. $x^2$ then what you are really saying is:
\begin{align}
x^2 - 1 &= n + nx^2 \\
\frac{x^2 - 1}{x^2} &= \frac{n + nx^2}{x^2} \\
\frac{x^2}{x^2} - \frac{1}{x^2} &= \frac{n}{x^2} + \frac{n^2x^2}{x^2}
\end{align}
I hope that clarifies your problem. 
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