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

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

Enrhifo

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

Camau Ateb

Defnyddio’r briodwedd ddosbarthu i luosi 1+\sqrt{2}+\sqrt{3} â 2+\sqrt{2}-\sqrt{6} a chyfuno termau tebyg.

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

Sgwâr \sqrt{2} yw 2.

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

Adio 2 a 2 i gael 4.

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

Ffactora 6=2\times 3. Ailysgrifennu ail isradd y lluoswm \sqrt{2\times 3} fel lluoswm ail israddau \sqrt{2}\sqrt{3}.

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

Lluosi \sqrt{2} a \sqrt{2} i gael 2.

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

Cyfuno -2\sqrt{3} a 2\sqrt{3} i gael 0.

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

I luosi \sqrt{3} a \sqrt{2}, dylid lluosi'r rhifau dan yr ail isradd.

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

Cyfuno -\sqrt{6} a \sqrt{6} i gael 0.

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

Ffactora 6=3\times 2. Ailysgrifennu ail isradd y lluoswm \sqrt{3\times 2} fel lluoswm ail israddau \sqrt{3}\sqrt{2}.

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

Lluosi \sqrt{3} a \sqrt{3} i gael 3.

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

Cyfuno 3\sqrt{2} a -3\sqrt{2} i gael 0.

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

Defnyddio'r theorem binomaidd \left(a-b\right)^{2}=a^{2}-2ab+b^{2} i ehangu'r \left(\sqrt{3}-1\right)^{2}.

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

Sgwâr \sqrt{3} yw 3.

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

Adio 3 a 1 i gael 4.

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

I ddod o hyd i wrthwyneb 4-2\sqrt{3}, dewch o hyd i wrthwyneb pob term.

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

Tynnu 4 o 4 i gael 0.

$$2\sqrt{3}$$