Réitigh frac{4+sqrt{5}}{4-sqrt{5}}+frac{4-sqrt{5}}{4+sqrt{5}}

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

Luacháil

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

Céimeanna gearra réitigh

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi 4+\sqrt{5} chun ainmneoir \frac{4+\sqrt{5}}{4-\sqrt{5}} a thiontú in uimhir chóimheasta.

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

Mar shampla \left(4-\sqrt{5}\right)\left(4+\sqrt{5}\right). Is féidir iolrúchán a athrú ó bhonn go dtí difríocht na gcearnóg ag úsáid na rialach seo: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Cearnóg 4. Cearnóg \sqrt{5}.

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

Dealaigh 5 ó 16 chun 11 a fháil.

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

Méadaigh 4+\sqrt{5} agus 4+\sqrt{5} chun \left(4+\sqrt{5}\right)^{2} a fháil.

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

Úsáid an teoirim dhéthéarmach \left(a+b\right)^{2}=a^{2}+2ab+b^{2} chun \left(4+\sqrt{5}\right)^{2} a leathnú.

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

Is é 5 uimhir chearnach \sqrt{5}.

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

Suimigh 16 agus 5 chun 21 a fháil.

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

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi 4-\sqrt{5} chun ainmneoir \frac{4-\sqrt{5}}{4+\sqrt{5}} a thiontú in uimhir chóimheasta.

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

Mar shampla \left(4+\sqrt{5}\right)\left(4-\sqrt{5}\right). Is féidir iolrúchán a athrú ó bhonn go dtí difríocht na gcearnóg ag úsáid na rialach seo: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Cearnóg 4. Cearnóg \sqrt{5}.

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

Dealaigh 5 ó 16 chun 11 a fháil.

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

Méadaigh 4-\sqrt{5} agus 4-\sqrt{5} chun \left(4-\sqrt{5}\right)^{2} a fháil.

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

Úsáid an teoirim dhéthéarmach \left(a-b\right)^{2}=a^{2}-2ab+b^{2} chun \left(4-\sqrt{5}\right)^{2} a leathnú.

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

Is é 5 uimhir chearnach \sqrt{5}.

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

Suimigh 16 agus 5 chun 21 a fháil.

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

Tá an t-ainmneoir céanna ag \frac{21+8\sqrt{5}}{11} agus \frac{21-8\sqrt{5}}{11} agus, mar sin, is féidir iad a shuimiú trína n-uimhreoirí a shuimiú.

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

Déan áirimh in 21+8\sqrt{5}+21-8\sqrt{5}.

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

Fachtóirigh

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