Thread: Operations on Real Numbers and Rational Equations

1. Operations on Real Numbers and Rational Equations

$[(x^2n + x^n)/(2x - 2)] / [(4x^n + 4)/(x^n+1 - x^n)]$

NOTE: $(x^2n + x^n)$ : The power isn't 2, it's 2n.
$(x^n+1 - x^n)$ : The power isn't n, it's n+1.

Sorry, I don't know how to get it to work properly.

$[(x-1)/(x+1) - (x+1)/(x-1)] / [(x-1)/(x+1) + (x+1)/(x-1)]$

Last two problems.

2. [(x^2n + x^n)/(2x - 2)] / [(4x^n + 4)/(x^n+1 - x^n)]
$\frac {\frac {x^{2n}+x^n}{2x-2}}{\frac {4x^n+4}{x^{n+1}-x^n}}$

to get more than one thingy in a superscript, surround them with curly brackets eg x^{2n}.

Just factorise all of the brackets then cancel. Remember that $x^{2n} = x^n*x^n$ and $\frac {a/b}{c/d} = \frac {ad}{bc}$

3. $
\frac{{\frac{{x^{2n} + x^n }}
{{2x - 2}}}}
{{\frac{{4x^n + 4}}
{{x^{n + 1} - x^n }}}} = \frac{{x^{2n} + x^n }}
{{2x - 2}} \cdot \frac{{x^{n + 1} - x^n }}
{{4x^n + 4}} = \frac{{x^n \left( {x^n + 1} \right)}}
{{2\left( {x - 1} \right)}} \cdot \frac{{x^n \left( {x - 1} \right)}}
{{4\left( {x^n + 1} \right)}} = \frac{{x^n x^n }}
{{\left( 2 \right)\left( 4 \right)}} = \frac{{x^{2n} }}
{8}
$

$
\boxed{
{\text{Let }}a = x - 1\text{;
}
{\text{ }}b = x + 1 }
$
:

$\frac{{\frac{a}
{b} - \frac{b}
{a}}}
{{\frac{a}
{b} + \frac{b}
{a}}} = \frac{{\frac{{a^2 - b^2 }}
{{ab}}}}
{{\frac{{a^2 + b^2 }}
{{ab}}}}$
$= \frac{{a^2 - b^2 }}
{{ab}} \cdot \frac{{ab}}
{{a^2 + b^2 }} = \frac{{a^2 - b^2 }}
{{a^2 + b^2 }} = \frac{{\left( {a + b} \right)\left( {a - b} \right)}}
{{a^2 + b^2 }}$

$\frac{{\left( {a + b} \right)\left( {a - b} \right)}}
{{a^2 + b^2 }} =$
$\frac{{\left( {\left( {x - 1} \right) + \left( {x + 1} \right)} \right)\left( {\left( {x - 1} \right) - \left( {x + 1} \right)} \right)}}
{{\left( {x - 1} \right)^2 + \left( {x + 1} \right)^2 }} =$
$\frac{{\left( {2x} \right)\left( { - 2} \right)}}
{{\left( {x^2 - 2x + 1} \right) + \left( {x^2 + 2x + 1} \right)}} =$
$\frac{{ - 4x}}
{{2x^2 + 2}} = \frac{{ - 2x}}
{{x^2 + 1}}$