# How do you simplify x^(3/2)((x+x^5)/(2-x^2))?

Jan 14, 2018

$\frac{{x}^{\frac{3}{2}} \left(x + {x}^{5}\right)}{2 - {x}^{2}}$, or $\frac{{x}^{\frac{5}{2}} \left(1 + {x}^{4}\right)}{2 - {x}^{2}}$

#### Explanation:

${x}^{\frac{3}{2}} \left(\frac{x + {x}^{5}}{2 - {x}^{2}}\right)$

${x}^{\frac{3}{2}} = {x}^{3} \cdot {x}^{\frac{1}{2}}$

${x}^{\frac{1}{2}} = \sqrt{x} \to {x}^{\frac{3}{2}} = \sqrt{{x}^{3}}$ or $x \sqrt{x}$

${x}^{\frac{3}{2}} \cdot \left(\frac{x + {x}^{5}}{2 - {x}^{2}}\right) = \frac{{x}^{\frac{3}{2}} \left(x + {x}^{5}\right)}{2 - {x}^{2}}$

${x}^{\frac{3}{2}} \left(x + {x}^{5}\right) = \left({x}^{\frac{3}{2}} \cdot {x}^{1}\right) + \left({x}^{\frac{3}{2}} \cdot {x}^{5}\right)$

${x}^{\frac{3}{2}} \cdot {x}^{1} = {x}^{\frac{3}{2} + 1} = {x}^{\frac{5}{2}}$
${x}^{\frac{3}{2}} \cdot {x}^{5} = {x}^{\frac{3}{2} + 5} = {x}^{\frac{13}{2}}$

$\frac{{x}^{\frac{3}{2}} \left(x + {x}^{5}\right)}{2 - {x}^{2}} = \frac{{x}^{\frac{5}{2}} + {x}^{\frac{13}{2}}}{2 - {x}^{2}}$

${x}^{\frac{5}{2}} + {x}^{\frac{13}{2}} = {x}^{\frac{5}{2}} \left(1 + {x}^{\frac{8}{2}}\right)$
$= {x}^{\frac{5}{2}} \left(1 + {x}^{4}\right)$

the expression cannot be simplified further, so is either $\frac{{x}^{\frac{3}{2}} \left(x + {x}^{5}\right)}{2 - {x}^{2}}$, or $\frac{{x}^{\frac{5}{2}} \left(1 + {x}^{4}\right)}{2 - {x}^{2}}$