How do you simplify ((10ab)/(x² - y²)) / ((5a(x - y))/(3ax(x + y))?

May 8, 2018

$\frac{6 a b x}{x - y} ^ 2$

Explanation:

Let us check which term can be simplified further:
${x}^{2} - {y}^{2}$ = $\left(x + y\right) \left(x - y\right)$

Then we know: $\frac{1}{\frac{a}{b}}$ = $1 \times \frac{b}{a}$
so now we simplify the original expression:

$\frac{\frac{10 a b}{{x}^{2} - {y}^{2}}}{\frac{5 a \left(x - y\right)}{3 a x \left(x + y\right)}}$ = $\frac{10 a b}{\left(x + y\right) \left(x - y\right)} \times \frac{3 a x \left(x + y\right)}{5 a \left(x - y\right)}$

$\frac{\frac{10 a b}{{x}^{2} - {y}^{2}}}{\frac{5 a \left(x - y\right)}{3 a x \left(x + y\right)}}$ = $\frac{10 a b}{\cancel{\left(x + y\right)} \left(x - y\right)} \times \frac{3 a x \cancel{\left(x + y\right)}}{5 a \left(x - y\right)}$

$\frac{\frac{10 a b}{{x}^{2} - {y}^{2}}}{\frac{5 a \left(x - y\right)}{3 a x \left(x + y\right)}}$ = (10ab)/((x-y))xx(3ax)/(5a(x-y)

$\frac{\frac{10 a b}{{x}^{2} - {y}^{2}}}{\frac{5 a \left(x - y\right)}{3 a x \left(x + y\right)}}$ = $\frac{30 {a}^{2} b x}{\left(5 a\right) {\left(x - y\right)}^{2}}$

$\frac{\frac{10 a b}{{x}^{2} - {y}^{2}}}{\frac{5 a \left(x - y\right)}{3 a x \left(x + y\right)}}$ = $\left(\frac{30}{5}\right) \times \frac{\left({a}^{2 - 1} \times b x\right)}{x - y} ^ 2$

$\frac{\frac{10 a b}{{x}^{2} - {y}^{2}}}{\frac{5 a \left(x - y\right)}{3 a x \left(x + y\right)}}$ = $\frac{6 a b x}{x - y} ^ 2$