Example

3(r+2s)=2t-4
a⋅(b-2)=3b
3(r+2s)=2t-4
a⋅(b-2)=3b

Question

$$\frac{ { x }^{ 3 } }{ { x }^{ 3 } +2 { x }^{ 2 } +1 } \times \frac{ { x }^{ 4- } 2x }{ { x }^{ 4 } -2 }$$

Answer

$$(2*x^(7-+1))/((x^3+2*x^2+1)*(x^4-2))$$

Solution


Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[\frac{{x}^{3}}{{x}^{3}+2{x}^{2}+1}\times \frac{{x}^{4-+1}\times 2}{{x}^{4}-2}\]
Regroup terms.
\[\frac{{x}^{3}}{{x}^{3}+2{x}^{2}+1}\times \frac{2{x}^{4-+1}}{{x}^{4}-2}\]
Use this rule: \(\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}\).
\[\frac{{x}^{3}\times 2{x}^{4-+1}}{({x}^{3}+2{x}^{2}+1)({x}^{4}-2)}\]
Regroup terms.
\[\frac{2{x}^{3}{x}^{4-+1}}{({x}^{3}+2{x}^{2}+1)({x}^{4}-2)}\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[\frac{2{x}^{3+4-+1}}{({x}^{3}+2{x}^{2}+1)({x}^{4}-2)}\]
Simplify  \(3+4\)  to  \(7\).
\[\frac{2{x}^{7-+1}}{({x}^{3}+2{x}^{2}+1)({x}^{4}-2)}\]
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