Whakaoti i te frac{4+sqrt{5}}{4-sqrt{5}}+frac{4-sqrt{5}}{4+sqrt{5}}

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

Aromātai

$$\frac{42}{11}\approx 3.818181818$$

Ngā Hipanga Rongoā Poto

Whakangāwaritia te tauraro o \frac{4+\sqrt{5}}{4-\sqrt{5}} mā te whakarea i te taurunga me te tauraro ki te 4+\sqrt{5}.

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

Whakaarohia te \left(4-\sqrt{5}\right)\left(4+\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}.

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

Pūrua 4. Pūrua \sqrt{5}.

$$\frac{\left(4+\sqrt{5}\right)\left(4+\sqrt{5}\right)}{16-5}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$$

Tangohia te 5 i te 16, ka 11.

$$\frac{\left(4+\sqrt{5}\right)\left(4+\sqrt{5}\right)}{11}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$$

Whakareatia te 4+\sqrt{5} ki te 4+\sqrt{5}, ka \left(4+\sqrt{5}\right)^{2}.

$$\frac{\left(4+\sqrt{5}\right)^{2}}{11}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(4+\sqrt{5}\right)^{2}.

$$\frac{16+8\sqrt{5}+\left(\sqrt{5}\right)^{2}}{11}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$$

Ko te pūrua o \sqrt{5} ko 5.

$$\frac{16+8\sqrt{5}+5}{11}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$$

Tāpirihia te 16 ki te 5, ka 21.

$$\frac{21+8\sqrt{5}}{11}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$$

Whakangāwaritia te tauraro o \frac{4-\sqrt{5}}{4+\sqrt{5}} mā te whakarea i te taurunga me te tauraro ki te 4-\sqrt{5}.

$$\frac{21+8\sqrt{5}}{11}+\frac{\left(4-\sqrt{5}\right)\left(4-\sqrt{5}\right)}{\left(4+\sqrt{5}\right)\left(4-\sqrt{5}\right)}$$

Whakaarohia te \left(4+\sqrt{5}\right)\left(4-\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}.

$$\frac{21+8\sqrt{5}}{11}+\frac{\left(4-\sqrt{5}\right)\left(4-\sqrt{5}\right)}{4^{2}-\left(\sqrt{5}\right)^{2}}$$

Pūrua 4. Pūrua \sqrt{5}.

$$\frac{21+8\sqrt{5}}{11}+\frac{\left(4-\sqrt{5}\right)\left(4-\sqrt{5}\right)}{16-5}$$

Tangohia te 5 i te 16, ka 11.

$$\frac{21+8\sqrt{5}}{11}+\frac{\left(4-\sqrt{5}\right)\left(4-\sqrt{5}\right)}{11}$$

Whakareatia te 4-\sqrt{5} ki te 4-\sqrt{5}, ka \left(4-\sqrt{5}\right)^{2}.

$$\frac{21+8\sqrt{5}}{11}+\frac{\left(4-\sqrt{5}\right)^{2}}{11}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(4-\sqrt{5}\right)^{2}.

$$\frac{21+8\sqrt{5}}{11}+\frac{16-8\sqrt{5}+\left(\sqrt{5}\right)^{2}}{11}$$

Ko te pūrua o \sqrt{5} ko 5.

$$\frac{21+8\sqrt{5}}{11}+\frac{16-8\sqrt{5}+5}{11}$$

Tāpirihia te 16 ki te 5, ka 21.

$$\frac{21+8\sqrt{5}}{11}+\frac{21-8\sqrt{5}}{11}$$

Tā te mea he rite te tauraro o \frac{21+8\sqrt{5}}{11} me \frac{21-8\sqrt{5}}{11}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{21+8\sqrt{5}+21-8\sqrt{5}}{11}$$

Mahia ngā tātaitai i roto o 21+8\sqrt{5}+21-8\sqrt{5}.

$$\frac{42}{11}$$

Tauwehe

$$\frac{2 \cdot 3 \cdot 7}{11} = 3\frac{9}{11} = 3.8181818181818183$$