Riješi 2mn/m^3+n^3+2m/m^2-n^2-1/m-n

$$\frac { 2 m n } { m ^ { 3 } + n ^ { 3 } } + \frac { 2 m } { m ^ { 2 } - n ^ { 2 } } - \frac { 1 } { m - n }$$

Izračunaj

$$\frac{m^{2}+mn+n^{2}}{m^{3}+n^{3}}$$

Kratki koraci za rješenje

Rastavite m^{3}+n^{3} na faktore. Rastavite m^{2}-n^{2} na faktore.

$$\frac{2mn}{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}+\frac{2m}{\left(m+n\right)\left(m-n\right)}-\frac{1}{m-n}$$

Da biste zbrojili ili oduzeli izraze, proširite ih da bi imali iste nazivnike. Najmanji zajednički višekratnik brojeva \left(m+n\right)\left(m^{2}-mn+n^{2}\right) i \left(m+n\right)\left(m-n\right) jest \left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right). Pomnožite \frac{2mn}{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)} i \frac{m-n}{m-n}. Pomnožite \frac{2m}{\left(m+n\right)\left(m-n\right)} i \frac{m^{2}-mn+n^{2}}{m^{2}-mn+n^{2}}.

$$\frac{2mn\left(m-n\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}+\frac{2m\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}-\frac{1}{m-n}$$

Budući da \frac{2mn\left(m-n\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)} i \frac{2m\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)} imaju isti nazivnik, zbrojite ih zbrajanjem njihovih brojnika.

$$\frac{2mn\left(m-n\right)+2m\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}-\frac{1}{m-n}$$

Pomnožite izraz 2mn\left(m-n\right)+2m\left(m^{2}-mn+n^{2}\right).

$$\frac{2m^{2}n-2mn^{2}+2m^{3}-2m^{2}n+2mn^{2}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}-\frac{1}{m-n}$$

Kombinirajte slične izraze u 2m^{2}n-2mn^{2}+2m^{3}-2m^{2}n+2mn^{2}.

$$\frac{2m^{3}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}-\frac{1}{m-n}$$

Da biste zbrojili ili oduzeli izraze, proširite ih da bi imali iste nazivnike. Najmanji zajednički višekratnik brojeva \left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right) i m-n jest \left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right). Pomnožite \frac{1}{m-n} i \frac{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}.

$$\frac{2m^{3}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}-\frac{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}$$

Budući da \frac{2m^{3}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)} i \frac{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)} imaju isti nazivnik, oduzmite ih oduzimanje njihovih brojnika.

$$\frac{2m^{3}-\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}$$

Pomnožite izraz 2m^{3}-\left(m+n\right)\left(m^{2}-mn+n^{2}\right).

$$\frac{2m^{3}-m^{3}+m^{2}n-mn^{2}-nm^{2}+n^{2}m-n^{3}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}$$

Kombinirajte slične izraze u 2m^{3}-m^{3}+m^{2}n-mn^{2}-nm^{2}+n^{2}m-n^{3}.

$$\frac{m^{3}-n^{3}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}$$

Rastavite na faktore izraze koji još nisu rastavljeni na faktore u izrazu \frac{m^{3}-n^{3}}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}.

$$\frac{\left(m-n\right)\left(m^{2}+mn+n^{2}\right)}{\left(m+n\right)\left(m-n\right)\left(m^{2}-mn+n^{2}\right)}$$

Skratite m-n u brojniku i nazivniku.

$$\frac{m^{2}+mn+n^{2}}{\left(m+n\right)\left(m^{2}-mn+n^{2}\right)}$$

Proširivanje broja \left(m+n\right)\left(m^{2}-mn+n^{2}\right).

$$\frac{m^{2}+mn+n^{2}}{m^{3}+n^{3}}$$

Diferenciraj u odnosu na m

$$\frac{-m^{4}+2mn^{3}+n^{4}-2nm^{3}-3\left(mn\right)^{2}}{\left(m^{3}+n^{3}\right)^{2}}$$