Whakaoti i te frac{2+sqrt{5}}{2-sqrt{5}}

$$\frac { 2 + \sqrt { 5 } } { 2 - \sqrt { 5 } }$$

Aromātai

$$-4\sqrt{5}-9\approx -17.94427191$$

Ngā Upane Otinga

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

$$\frac{\left(2+\sqrt{5}\right)\left(2+\sqrt{5}\right)}{\left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right)}$$

Whakaarohia te \left(2-\sqrt{5}\right)\left(2+\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}.

$$\frac{\left(2+\sqrt{5}\right)\left(2+\sqrt{5}\right)}{2^{2}-\left(\sqrt{5}\right)^{2}}$$

Pūrua 2. Pūrua \sqrt{5}.

$$\frac{\left(2+\sqrt{5}\right)\left(2+\sqrt{5}\right)}{4-5}$$

Tangohia te 5 i te 4, ka -1.

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

Whakareatia te 2+\sqrt{5} ki te 2+\sqrt{5}, ka \left(2+\sqrt{5}\right)^{2}.

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

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(2+\sqrt{5}\right)^{2}.

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

Ko te pūrua o \sqrt{5} ko 5.

$$\frac{4+4\sqrt{5}+5}{-1}$$

Tāpirihia te 4 ki te 5, ka 9.

$$\frac{9+4\sqrt{5}}{-1}$$

Ko te mea whakawehea ki te -1 ka hōmai i tōna kōaro. Hei kimi i te tauaro o 9+4\sqrt{5}, kimihia te tauaro o ia taurangi.

$$-9-4\sqrt{5}$$