Whakaoti i te frac{8+4-2sqrt{5}-4sqrt{5}+2sqrt{10}}{1-sqrt{5}}

$$\frac { 8 + 4 - 2 \sqrt { 5 } - 4 \sqrt { 5 } + 2 \sqrt { 10 } } { 1 - \sqrt { 5 } }$$

Aromātai

$$\frac{9-\sqrt{10}-3\sqrt{5}-5\sqrt{2}}{2}\approx -3.970774702$$

Ngā Upane Otinga

Tāpirihia te 8 ki te 4, ka 12.

$$\frac{12-2\sqrt{5}-4\sqrt{5}+2\sqrt{10}}{1-\sqrt{5}}$$

Pahekotia te -2\sqrt{5} me -4\sqrt{5}, ka -6\sqrt{5}.

$$\frac{12-6\sqrt{5}+2\sqrt{10}}{1-\sqrt{5}}$$

Whakangāwaritia te tauraro o \frac{12-6\sqrt{5}+2\sqrt{10}}{1-\sqrt{5}} mā te whakarea i te taurunga me te tauraro ki te 1+\sqrt{5}.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)}$$

Whakaarohia te \left(1-\sqrt{5}\right)\left(1+\sqrt{5}\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}.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{1^{2}-\left(\sqrt{5}\right)^{2}}$$

Pūrua 1. Pūrua \sqrt{5}.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{1-5}$$

Tangohia te 5 i te 1, ka -4.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{-4}$$

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o 12-6\sqrt{5}+2\sqrt{10} ki ia tau o 1+\sqrt{5}.

$$\frac{12+12\sqrt{5}-6\sqrt{5}-6\left(\sqrt{5}\right)^{2}+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Pahekotia te 12\sqrt{5} me -6\sqrt{5}, ka 6\sqrt{5}.

$$\frac{12+6\sqrt{5}-6\left(\sqrt{5}\right)^{2}+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Ko te pūrua o \sqrt{5} ko 5.

$$\frac{12+6\sqrt{5}-6\times 5+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Whakareatia te -6 ki te 5, ka -30.

$$\frac{12+6\sqrt{5}-30+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Tangohia te 30 i te 12, ka -18.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Tauwehea te 10=5\times 2. Tuhia anō te pūtake rua o te hua \sqrt{5\times 2} hei hua o ngā pūtake rua \sqrt{5}\sqrt{2}.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+2\sqrt{5}\sqrt{2}\sqrt{5}}{-4}$$

Whakareatia te \sqrt{5} ki te \sqrt{5}, ka 5.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+2\times 5\sqrt{2}}{-4}$$

Whakareatia te 2 ki te 5, ka 10.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+10\sqrt{2}}{-4}$$

Tauwehe

$$\frac{9 - \sqrt{10} - 3 \sqrt{5} - 5 \sqrt{2}}{2} = -3.9707747022666124$$