espandere α^2β^2(αβ+α+β+1)^2

Soluzione

espandere

α^{2} β^{2} (α β + α + β + 1)^{2}

α^{4} β^{4} + 2 α^{3} β^{4} + 2 α^{4} β^{3} + 4 α^{3} β^{3} + α^{4} β^{2} + 2 α^{3} β^{2} + α^{2} β^{4} + 2 α^{2} β^{3} + α^{2} β^{2}

Fasi della soluzione

α^{2} β^{2} (α β + α + β + 1)^{2}

Applicare la regola degli esponenti:

a^{m} b^{m} = (ab)^{m},

assumendo

a \geq 0, b \geq 0

α^{2} β^{2} (α β + α + β + 1)^{2} = (α β (α β + α + β + 1))^{2}

= (α β (α β + α + β + 1))^{2}

Espandi

α β (α β + α + β + 1) : α^{2} β^{2} + α^{2} β + α β^{2} + α β

= (α^{2} β^{2} + α^{2} β + α β^{2} + α β)^{2}

(α^{2} β^{2} + α^{2} β + α β^{2} + α β)^{2} = (α^{2} β^{2} + α^{2} β + α β^{2} + α β) (α^{2} β^{2} + α^{2} β + α β^{2} + α β)

= (α^{2} β^{2} + α^{2} β + α β^{2} + α β) (α^{2} β^{2} + α^{2} β + α β^{2} + α β)

Espandi

(α^{2} β^{2} + α^{2} β + α β^{2} + α β) (α^{2} β^{2} + α^{2} β + α β^{2} + α β) : α^{4} β^{4} + 2 α^{3} β^{4} + 2 α^{4} β^{3} + 4 α^{3} β^{3} + α^{4} β^{2} + 2 α^{3} β^{2} + α^{2} β^{4} + 2 α^{2} β^{3} + α^{2} β^{2}

= α^{4} β^{4} + 2 α^{3} β^{4} + 2 α^{4} β^{3} + 4 α^{3} β^{3} + α^{4} β^{2} + 2 α^{3} β^{2} + α^{2} β^{4} + 2 α^{2} β^{3} + α^{2} β^{2}

e x p an d (x^{2} - 4) d x

\frac{d}{d x} ((0) (12 x^{3} - 24 x))

e x p an d 4 (e^{3 x} + 3 e^{3 x} x)

\frac{\partial}{\partial x} ((x y) (ey + 2 z))

e x p an d (3 x + y) d x + 2 x d y