Réitigh (1+sqrt{2}+sqrt{3})(2+sqrt{2}-sqrt{6})-(sqrt{3}-1)^2

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

Luacháil

$$2\sqrt{3}\approx 3.464101615$$

Céimeanna Réitigh

Úsáid an t-airí dáileach chun 1+\sqrt{2}+\sqrt{3} a mhéadú faoi 2+\sqrt{2}-\sqrt{6} agus chun téarmaí comhchosúla a chumasc.

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

Is é 2 uimhir chearnach \sqrt{2}.

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

Suimigh 2 agus 2 chun 4 a fháil.

$$4+3\sqrt{2}-\sqrt{6}-\sqrt{2}\sqrt{6}+2\sqrt{3}+\sqrt{3}\sqrt{2}-\sqrt{3}\sqrt{6}-\left(\sqrt{3}-1\right)^{2}$$

Fachtóirigh 6=2\times 3. Athscríobh fréamh cearnach an toraidh \sqrt{2\times 3} mar thoradh na bhfréamhacha cearnacha \sqrt{2}\sqrt{3}.

$$4+3\sqrt{2}-\sqrt{6}-\sqrt{2}\sqrt{2}\sqrt{3}+2\sqrt{3}+\sqrt{3}\sqrt{2}-\sqrt{3}\sqrt{6}-\left(\sqrt{3}-1\right)^{2}$$

Méadaigh \sqrt{2} agus \sqrt{2} chun 2 a fháil.

$$4+3\sqrt{2}-\sqrt{6}-2\sqrt{3}+2\sqrt{3}+\sqrt{3}\sqrt{2}-\sqrt{3}\sqrt{6}-\left(\sqrt{3}-1\right)^{2}$$

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

$$4+3\sqrt{2}-\sqrt{6}+\sqrt{3}\sqrt{2}-\sqrt{3}\sqrt{6}-\left(\sqrt{3}-1\right)^{2}$$

Iolraigh na huimhreacha faoin bhfréamh cearnach chun \sqrt{3} agus \sqrt{2} a iolrú.

$$4+3\sqrt{2}-\sqrt{6}+\sqrt{6}-\sqrt{3}\sqrt{6}-\left(\sqrt{3}-1\right)^{2}$$

Comhcheangail -\sqrt{6} agus \sqrt{6} chun 0 a fháil.

$$4+3\sqrt{2}-\sqrt{3}\sqrt{6}-\left(\sqrt{3}-1\right)^{2}$$

Fachtóirigh 6=3\times 2. Athscríobh fréamh cearnach an toraidh \sqrt{3\times 2} mar thoradh na bhfréamhacha cearnacha \sqrt{3}\sqrt{2}.

$$4+3\sqrt{2}-\sqrt{3}\sqrt{3}\sqrt{2}-\left(\sqrt{3}-1\right)^{2}$$

Méadaigh \sqrt{3} agus \sqrt{3} chun 3 a fháil.

$$4+3\sqrt{2}-3\sqrt{2}-\left(\sqrt{3}-1\right)^{2}$$

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

$$4-\left(\sqrt{3}-1\right)^{2}$$

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

$$4-\left(\left(\sqrt{3}\right)^{2}-2\sqrt{3}+1\right)$$

Is é 3 uimhir chearnach \sqrt{3}.

$$4-\left(3-2\sqrt{3}+1\right)$$

Suimigh 3 agus 1 chun 4 a fháil.

$$4-\left(4-2\sqrt{3}\right)$$

Chun an mhalairt ar 4-2\sqrt{3} a aimsiú, aimsigh an mhalairt ar gach téarma.

$$4-4+2\sqrt{3}$$

Dealaigh 4 ó 4 chun 0 a fháil.

$$2\sqrt{3}$$