Issolvi 3-frac{sqrt{2}}{(1-sqrt{5}}

$$3- \frac{ \sqrt{ 2 } }{ (1- \sqrt{ 5 } }$$

Evalwa

$$\frac{\sqrt{2}+\sqrt{10}+12}{4}\approx 4.144122806$$

Passi Qosra tas-Soluzzjonijiet

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

$$3-\frac{\sqrt{2}\left(1+\sqrt{5}\right)}{\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)}$$

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}.

$$3-\frac{\sqrt{2}\left(1+\sqrt{5}\right)}{1^{2}-\left(\sqrt{5}\right)^{2}}$$

Ikkwadra 1. Ikkwadra \sqrt{5}.

$$3-\frac{\sqrt{2}\left(1+\sqrt{5}\right)}{1-5}$$

Naqqas 5 minn 1 biex tikseb -4.

$$3-\frac{\sqrt{2}\left(1+\sqrt{5}\right)}{-4}$$

Uża l-propjetà distributtiva biex timmultiplika \sqrt{2} b'1+\sqrt{5}.

$$3-\frac{\sqrt{2}+\sqrt{2}\sqrt{5}}{-4}$$

Biex timmultiplika \sqrt{2} u \sqrt{5}, immultiplika n-numri taħt l-għerq kwadrat.

$$3-\frac{\sqrt{2}+\sqrt{10}}{-4}$$

Immultiplika kemm in-numeratur u kif ukoll id-denominatur b’-1.

$$3-\frac{-\sqrt{2}-\sqrt{10}}{4}$$

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. Immultiplika 3 b'\frac{4}{4}.

$$\frac{3\times 4}{4}-\frac{-\sqrt{2}-\sqrt{10}}{4}$$

Billi \frac{3\times 4}{4} u \frac{-\sqrt{2}-\sqrt{10}}{4} għandhom l-istess denominatur, naqqashom billi tnaqqas in-numeraturi tagħhom.

$$\frac{3\times 4-\left(-\sqrt{2}-\sqrt{10}\right)}{4}$$

Agħmel il-multiplikazzjonijiet fi 3\times 4-\left(-\sqrt{2}-\sqrt{10}\right).

$$\frac{12+\sqrt{2}+\sqrt{10}}{4}$$