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

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

Evalwa

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

Passi Qosra tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{4+\sqrt{5}}{4-\sqrt{5}} billi timmultiplika in-numeratur u d-denominatur mill-4+\sqrt{5}.

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

Ikkunsidra li \left(4-\sqrt{5}\right)\left(4+\sqrt{5}\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Ikkwadra 4. Ikkwadra \sqrt{5}.

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

Naqqas 5 minn 16 biex tikseb 11.

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

Immultiplika 4+\sqrt{5} u 4+\sqrt{5} biex tikseb \left(4+\sqrt{5}\right)^{2}.

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

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(4+\sqrt{5}\right)^{2}.

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

Il-kwadrat ta' \sqrt{5} huwa 5.

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

Żid 16 u 5 biex tikseb 21.

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

Irrazzjonalizza d-denominatur tal-\frac{4-\sqrt{5}}{4+\sqrt{5}} billi timmultiplika in-numeratur u d-denominatur mill-4-\sqrt{5}.

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

Ikkunsidra li \left(4+\sqrt{5}\right)\left(4-\sqrt{5}\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Ikkwadra 4. Ikkwadra \sqrt{5}.

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

Naqqas 5 minn 16 biex tikseb 11.

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

Immultiplika 4-\sqrt{5} u 4-\sqrt{5} biex tikseb \left(4-\sqrt{5}\right)^{2}.

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

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(4-\sqrt{5}\right)^{2}.

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

Il-kwadrat ta' \sqrt{5} huwa 5.

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

Żid 16 u 5 biex tikseb 21.

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

Billi \frac{21+8\sqrt{5}}{11} u \frac{21-8\sqrt{5}}{11} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

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

Agħmel il-kalkoli fi 21+8\sqrt{5}+21-8\sqrt{5}.

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

Fattur

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