Whakaoti i te (1+sqrt{2}+sqrt{3})(2+sqrt{2}-sqrt{6})-(sqrt{3}-1)^2

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

Aromātai

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

Ngā Upane Otinga

Whakamahia te āhuatanga tuaritanga hei whakarea te 1+\sqrt{2}+\sqrt{3} ki te 2+\sqrt{2}-\sqrt{6} ka whakakotahi i ngā kupu rite.

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

Ko te pūrua o \sqrt{2} ko 2.

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

Tāpirihia te 2 ki te 2, ka 4.

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

Tauwehea te 6=2\times 3. Tuhia anō te pūtake rua o te hua \sqrt{2\times 3} hei hua o ngā pūtake rua \sqrt{2}\sqrt{3}.

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

Whakareatia te \sqrt{2} ki te \sqrt{2}, ka 2.

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

Pahekotia te -2\sqrt{3} me 2\sqrt{3}, ka 0.

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

Hei whakarea \sqrt{3} me \sqrt{2}, whakareatia ngā tau i raro i te pūtake rua.

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

Pahekotia te -\sqrt{6} me \sqrt{6}, ka 0.

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

Tauwehea te 6=3\times 2. Tuhia anō te pūtake rua o te hua \sqrt{3\times 2} hei hua o ngā pūtake rua \sqrt{3}\sqrt{2}.

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

Whakareatia te \sqrt{3} ki te \sqrt{3}, ka 3.

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

Pahekotia te 3\sqrt{2} me -3\sqrt{2}, ka 0.

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

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\sqrt{3}-1\right)^{2}.

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

Ko te pūrua o \sqrt{3} ko 3.

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

Tāpirihia te 3 ki te 1, ka 4.

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

Hei kimi i te tauaro o 4-2\sqrt{3}, kimihia te tauaro o ia taurangi.

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

Tangohia te 4 i te 4, ka 0.

$$2\sqrt{3}$$