Question

1. Simplify the following rational expression in the lowest terms
$\left(\frac{a{x}^2-x^3}{a^3+x^3}\times \frac{a^2-ax+x^2}{a^2{x}^2+x^4}\right)÷\frac{a^2-2ax+x^2}{a^4-x^4}$

2. Simplify:-
$\left[1-\frac{4ax}{{\left(a+x\right)}^2}\right]\times \frac{{\left(x+a\right)}^2}{x^2-a^2}$

Answer 1

Dear student

$$\begin{gathered} \left[1-\frac{4\mathrm{ax}}{{\left(\mathrm{a}+\mathrm{x}\right)}^2}\right]\times \frac{{\left(\mathrm{x}+\mathrm{a}\right)}^2}{{\mathrm{x}}^2-{\mathrm{a}}^2} \\ =\left[\frac{{\left(\mathrm{a}+\mathrm{x}\right)}^2-4\mathrm{ax}}{\cancel{{\left(\mathrm{a}+\mathrm{x}\right)}^2}}\right]\times \frac{\cancel{{\left(\mathrm{x}+\mathrm{a}\right)}^2}}{{\mathrm{x}}^2-{\mathrm{a}}^2} \\ =\frac{{\mathrm{a}}^2+{\mathrm{x}}^2+2\mathrm{ax}-4\mathrm{ax}}{{\mathrm{x}}^2-{\mathrm{a}}^2} \\ =\frac{{\mathrm{a}}^2+{\mathrm{x}}^2-2\mathrm{ax}}{{\mathrm{x}}^2-{\mathrm{a}}^2} \\ =\frac{{\left(\mathrm{a}-\mathrm{x}\right)}^2}{{\mathrm{x}}^2-{\mathrm{a}}^2} \\ \therefore \boxed{\left[\mathbf{1}\mathbf{-}\frac{\mathbf{4}\mathbf{ax}}{{\left(\mathbf{a}\mathbf{+}\mathbf{x}\right)}^{\mathbf{2}}}\right]\mathbf{\times }\frac{{\left(\mathbf{x}\mathbf{+}\mathbf{a}\right)}^{\mathbf{2}}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}{\mathbf{a}}^{\mathbf{2}}}\mathbf{=}\frac{{\left(\mathbf{a}\mathbf{-}\mathbf{x}\right)}^{\mathbf{2}}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}{\mathbf{a}}^{\mathbf{2}}}} \end{gathered}$$
For remaining queries we request you to post them in separate threads to have rapid assistance from our experts.
Regards