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

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

Arvuta

$$\frac{42}{11}\approx 3,818181818$$

Lühike lahenduskäik

Ratsionaliseerige korrutades lugeja ja 4+\sqrt{5} nimetaja \frac{4+\sqrt{5}}{4-\sqrt{5}} nimetaja.

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

Mõelge valemile \left(4-\sqrt{5}\right)\left(4+\sqrt{5}\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Tõstke 4 ruutu. Tõstke \sqrt{5} ruutu.

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

Lahutage 5 väärtusest 16, et leida 11.

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

Korrutage 4+\sqrt{5} ja 4+\sqrt{5}, et leida \left(4+\sqrt{5}\right)^{2}.

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

Kasutage kaksliikme \left(4+\sqrt{5}\right)^{2} arendamiseks binoomvalemit \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

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

\sqrt{5} ruut on 5.

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

Liitke 16 ja 5, et leida 21.

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

Ratsionaliseerige korrutades lugeja ja 4-\sqrt{5} nimetaja \frac{4-\sqrt{5}}{4+\sqrt{5}} nimetaja.

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

Mõelge valemile \left(4+\sqrt{5}\right)\left(4-\sqrt{5}\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Tõstke 4 ruutu. Tõstke \sqrt{5} ruutu.

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

Lahutage 5 väärtusest 16, et leida 11.

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

Korrutage 4-\sqrt{5} ja 4-\sqrt{5}, et leida \left(4-\sqrt{5}\right)^{2}.

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

Kasutage kaksliikme \left(4-\sqrt{5}\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

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

\sqrt{5} ruut on 5.

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

Liitke 16 ja 5, et leida 21.

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

Kuna murdudel \frac{21+8\sqrt{5}}{11} ja \frac{21-8\sqrt{5}}{11} on sama nimetaja, liitke nende lugejad.

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

Tehke arvutustehted avaldises 21+8\sqrt{5}+21-8\sqrt{5}.

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

Lahuta teguriteks

$$\frac{2 \cdot 3 \cdot 7}{11} = 3\frac{9}{11} = 3,8181818181818183$$