simplify xm+2nx^{m+2n} + x3m−8nx^{3m-8n} / x5m−6nx^{5m-6n}

asked by guest
on Nov 22, 2024 at 7:49 am



You asked:

Simplify the expression: xm+2n+x3m−8nx5m−6n{x}^{m + 2 n} + \frac{{x}^{3 m - 8 n}}{{x}^{5 m - 6 n}}

MathBot Answer:

Simplified:

xm+2n+x3m−8nx5m−6n=x−2m−2n+xm+2n {x}^{m + 2 n} + \frac{{x}^{3 m - 8 n}}{{x}^{5 m - 6 n}} = x^{- 2 m - 2 n} + x^{m + 2 n}

xm+2n+x3m−8nx5m−6n=xm+2n+x−5m+6nx3m−8n=x−5m+6nx3m−8n+x2nxm=x6nx3m−8nx5m+x2nxm=x6nx−8n+3mx5m+x2nxm=x6nx3mx5mx8n+x2nxm=1x2mx2n+x2nxm=x−2m−2n+xm+2n\begin{aligned}x^{m + 2 n} + \frac{x^{3 m - 8 n}}{x^{5 m - 6 n}}&=x^{m + 2 n} + x^{- 5 m + 6 n} x^{3 m - 8 n}\\&=x^{- 5 m + 6 n} x^{3 m - 8 n} + x^{2 n} x^{m}\\&=\frac{x^{6 n} x^{3 m - 8 n}}{x^{5 m}} + x^{2 n} x^{m}\\&=\frac{x^{6 n} x^{- 8 n + 3 m}}{x^{5 m}} + x^{2 n} x^{m}\\&=\frac{x^{6 n} x^{3 m}}{x^{5 m} x^{8 n}} + x^{2 n} x^{m}\\&=\frac{1}{x^{2 m} x^{2 n}} + x^{2 n} x^{m}\\&=x^{- 2 m - 2 n} + x^{m + 2 n}\end{aligned}

Expanded:

xm+2n+x3m−8nx5m−6n=x2nxm+x−2mx−2n {x}^{m + 2 n} + \frac{{x}^{3 m - 8 n}}{{x}^{5 m - 6 n}} = x^{2 n} x^{m} + x^{- 2 m} x^{- 2 n}

Factored:

xm+2n+x3m−8nx5m−6n=x2nxm+x−2mx−2n {x}^{m + 2 n} + \frac{{x}^{3 m - 8 n}}{{x}^{5 m - 6 n}} = x^{2 n} x^{m} + x^{- 2 m} x^{- 2 n}


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