Whakaoti i te frac{sqrt{12}+sqrt{6}+sqrt{2}+2}{sqrt{3}+1}

$$\frac { \sqrt { 12 } + \sqrt { 6 } + \sqrt { 2 } + 2 } { \sqrt { 3 } + 1 }$$

Aromātai

$$\sqrt{2}+2\approx 3.414213562$$

Ngā Upane Otinga

Tauwehea te 12=2^{2}\times 3. Tuhia anō te pūtake rua o te hua \sqrt{2^{2}\times 3} hei hua o ngā pūtake rua \sqrt{2^{2}}\sqrt{3}. Tuhia te pūtakerua o te 2^{2}.

$$\frac{2\sqrt{3}+\sqrt{6}+\sqrt{2}+2}{\sqrt{3}+1}$$

Whakangāwaritia te tauraro o \frac{2\sqrt{3}+\sqrt{6}+\sqrt{2}+2}{\sqrt{3}+1} mā te whakarea i te taurunga me te tauraro ki te \sqrt{3}-1.

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

Whakaarohia te \left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Pūrua \sqrt{3}. Pūrua 1.

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

Tangohia te 1 i te 3, ka 2.

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

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o 2\sqrt{3}+\sqrt{6}+\sqrt{2}+2 ki ia tau o \sqrt{3}-1.

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

Ko te pūrua o \sqrt{3} ko 3.

$$\frac{2\times 3-2\sqrt{3}+\sqrt{6}\sqrt{3}-\sqrt{6}+\sqrt{2}\sqrt{3}-\sqrt{2}+2\sqrt{3}-2}{2}$$

Whakareatia te 2 ki te 3, ka 6.

$$\frac{6-2\sqrt{3}+\sqrt{6}\sqrt{3}-\sqrt{6}+\sqrt{2}\sqrt{3}-\sqrt{2}+2\sqrt{3}-2}{2}$$

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}.

$$\frac{6-2\sqrt{3}+\sqrt{3}\sqrt{2}\sqrt{3}-\sqrt{6}+\sqrt{2}\sqrt{3}-\sqrt{2}+2\sqrt{3}-2}{2}$$

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

$$\frac{6-2\sqrt{3}+3\sqrt{2}-\sqrt{6}+\sqrt{2}\sqrt{3}-\sqrt{2}+2\sqrt{3}-2}{2}$$

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

$$\frac{6-2\sqrt{3}+3\sqrt{2}-\sqrt{6}+\sqrt{6}-\sqrt{2}+2\sqrt{3}-2}{2}$$

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

$$\frac{6-2\sqrt{3}+3\sqrt{2}-\sqrt{2}+2\sqrt{3}-2}{2}$$

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

$$\frac{6-2\sqrt{3}+2\sqrt{2}+2\sqrt{3}-2}{2}$$

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

$$\frac{6+2\sqrt{2}-2}{2}$$

Tangohia te 2 i te 6, ka 4.

$$\frac{4+2\sqrt{2}}{2}$$

Whakawehea ia wā o 4+2\sqrt{2} ki te 2, kia riro ko 2+\sqrt{2}.

$$2+\sqrt{2}$$