Issolvi frac{8+4-2sqrt{5}-4sqrt{5}+2sqrt{10}}{1-sqrt{5}}

$$\frac { 8 + 4 - 2 \sqrt { 5 } - 4 \sqrt { 5 } + 2 \sqrt { 10 } } { 1 - \sqrt { 5 } }$$

Evalwa

$$\frac{9-\sqrt{10}-3\sqrt{5}-5\sqrt{2}}{2}\approx -3.970774702$$

Passi tas-Soluzzjonijiet

Żid 8 u 4 biex tikseb 12.

$$\frac{12-2\sqrt{5}-4\sqrt{5}+2\sqrt{10}}{1-\sqrt{5}}$$

Ikkombina -2\sqrt{5} u -4\sqrt{5} biex tikseb -6\sqrt{5}.

$$\frac{12-6\sqrt{5}+2\sqrt{10}}{1-\sqrt{5}}$$

Irrazzjonalizza d-denominatur tal-\frac{12-6\sqrt{5}+2\sqrt{10}}{1-\sqrt{5}} billi timmultiplika in-numeratur u d-denominatur mill-1+\sqrt{5}.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)}$$

Ikkunsidra li \left(1-\sqrt{5}\right)\left(1+\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}.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{1^{2}-\left(\sqrt{5}\right)^{2}}$$

Ikkwadra 1. Ikkwadra \sqrt{5}.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{1-5}$$

Naqqas 5 minn 1 biex tikseb -4.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{-4}$$

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 12-6\sqrt{5}+2\sqrt{10} b'kull terminu ta' 1+\sqrt{5}.

$$\frac{12+12\sqrt{5}-6\sqrt{5}-6\left(\sqrt{5}\right)^{2}+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Ikkombina 12\sqrt{5} u -6\sqrt{5} biex tikseb 6\sqrt{5}.

$$\frac{12+6\sqrt{5}-6\left(\sqrt{5}\right)^{2}+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

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

$$\frac{12+6\sqrt{5}-6\times 5+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Immultiplika -6 u 5 biex tikseb -30.

$$\frac{12+6\sqrt{5}-30+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Naqqas 30 minn 12 biex tikseb -18.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Iffattura 10=5\times 2. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{5\times 2} bħala l-prodott tal-għeruq kwadrati \sqrt{5}\sqrt{2}.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+2\sqrt{5}\sqrt{2}\sqrt{5}}{-4}$$

Immultiplika \sqrt{5} u \sqrt{5} biex tikseb 5.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+2\times 5\sqrt{2}}{-4}$$

Immultiplika 2 u 5 biex tikseb 10.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+10\sqrt{2}}{-4}$$

Fattur

$$\frac{9 - \sqrt{10} - 3 \sqrt{5} - 5 \sqrt{2}}{2} = -3.9707747022666124$$