Réitigh (sqrt{2}+1)^3-(sqrt{2}-1)^3

$$( \sqrt { 2 } + 1 ) ^ { 3 } - ( \sqrt { 2 } - 1 ) ^ { 3 }$$

Luacháil

$$14$$

Céimeanna Réitigh

Úsáid an teoirim dhéthéarmach \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} chun \left(\sqrt{2}+1\right)^{3} a leathnú.

$$\left(\sqrt{2}\right)^{3}+3\left(\sqrt{2}\right)^{2}+3\sqrt{2}+1-\left(\sqrt{2}-1\right)^{3}$$

Is é 2 uimhir chearnach \sqrt{2}.

$$\left(\sqrt{2}\right)^{3}+3\times 2+3\sqrt{2}+1-\left(\sqrt{2}-1\right)^{3}$$

Méadaigh 3 agus 2 chun 6 a fháil.

$$\left(\sqrt{2}\right)^{3}+6+3\sqrt{2}+1-\left(\sqrt{2}-1\right)^{3}$$

Suimigh 6 agus 1 chun 7 a fháil.

$$\left(\sqrt{2}\right)^{3}+7+3\sqrt{2}-\left(\sqrt{2}-1\right)^{3}$$

Úsáid an teoirim dhéthéarmach \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} chun \left(\sqrt{2}-1\right)^{3} a leathnú.

$$\left(\sqrt{2}\right)^{3}+7+3\sqrt{2}-\left(\left(\sqrt{2}\right)^{3}-3\left(\sqrt{2}\right)^{2}+3\sqrt{2}-1\right)$$

Is é 2 uimhir chearnach \sqrt{2}.

$$\left(\sqrt{2}\right)^{3}+7+3\sqrt{2}-\left(\left(\sqrt{2}\right)^{3}-3\times 2+3\sqrt{2}-1\right)$$

Méadaigh -3 agus 2 chun -6 a fháil.

$$\left(\sqrt{2}\right)^{3}+7+3\sqrt{2}-\left(\left(\sqrt{2}\right)^{3}-6+3\sqrt{2}-1\right)$$

Dealaigh 1 ó -6 chun -7 a fháil.

$$\left(\sqrt{2}\right)^{3}+7+3\sqrt{2}-\left(\left(\sqrt{2}\right)^{3}-7+3\sqrt{2}\right)$$

Chun an mhalairt ar \left(\sqrt{2}\right)^{3}-7+3\sqrt{2} a aimsiú, aimsigh an mhalairt ar gach téarma.

$$\left(\sqrt{2}\right)^{3}+7+3\sqrt{2}-\left(\sqrt{2}\right)^{3}+7-3\sqrt{2}$$

Comhcheangail \left(\sqrt{2}\right)^{3} agus -\left(\sqrt{2}\right)^{3} chun 0 a fháil.

$$7+3\sqrt{2}+7-3\sqrt{2}$$

Suimigh 7 agus 7 chun 14 a fháil.

$$14+3\sqrt{2}-3\sqrt{2}$$

Comhcheangail 3\sqrt{2} agus -3\sqrt{2} chun 0 a fháil.

$$14$$

Fachtóirigh

$$2\times 7$$