# How do you simplify  a/(r+a) - a/(r-a)?

Nov 9, 2017

$- \frac{2 {a}^{2}}{\left(r + a\right) \left(r - a\right)}$

#### Explanation:

$\text{before we can subtract the fractions we require them to}$
$\text{have a "color(blue)"common denominator}$

$\text{multiply numerator/denominator of " a/(r+a)" by } \left(r - a\right)$

$\text{multiply numerator/denominator of "a/(r-a)" by } \left(r + a\right)$

$\Rightarrow \frac{a \left(r - a\right)}{\left(r + a\right) \left(r - a\right)} - \frac{a \left(r + a\right)}{\left(r + a\right) \left(r - a\right)}$

$\text{we now have a common denominator so can subtract}$
$\text{the numerators leaving the denominator}$

$= \frac{\cancel{a r} - {a}^{2} \cancel{- a r} - {a}^{2}}{\left(r + a\right) \left(r - a\right)}$

$= - \frac{2 {a}^{2}}{\left(r + a\right) \left(r - a\right)} \to \left(r \ne \pm a\right)$

Nov 9, 2017

See a solution process below:

#### Explanation:

First, we will put the fractions over common denominators by multiplying each fraction by the appropriate form of $1$:

$\left(\frac{r - a}{r - a} \times \frac{a}{r + a}\right) - \left(\frac{r + a}{r + a} \times \frac{a}{r - a}\right)$

$\frac{\left(r - a\right) a}{\left(r - a\right) \left(r + a\right)} - \frac{\left(r + a\right) a}{\left(r + a\right) \left(r - a\right)}$

$\frac{a r - {a}^{2}}{{r}^{2} + a r - a r - {a}^{2}} - \frac{a r + {a}^{2}}{{r}^{2} + a r - a r - {a}^{2}}$

$\frac{a r - {a}^{2}}{{r}^{2} - {a}^{2}} - \frac{a r + {a}^{2}}{{r}^{2} - {a}^{2}}$

We can now subtract the numerators over the common denominators:

$\frac{\left(a r - {a}^{2}\right) - \left(a r + {a}^{2}\right)}{{r}^{2} - {a}^{2}}$

$\frac{a r - {a}^{2} - a r - {a}^{2}}{{r}^{2} - {a}^{2}}$

$\frac{a r - a r - {a}^{2} - {a}^{2}}{{r}^{2} - {a}^{2}}$

$\frac{0 - 2 {a}^{2}}{{r}^{2} - {a}^{2}}$

$\frac{- 2 {a}^{2}}{{r}^{2} - {a}^{2}}$

However, we must look at the original expression and qualify the answer with:

Where $r + a \ne 0$ or $r - a \ne 0$