Issolvi frac{sqrt{14}+2}{1-sqrt{7}}

$$\frac{ \sqrt{ 14 } +2 }{ 1- \sqrt{ 7 } }$$

Evalwa

$$-\frac{\sqrt{7}}{3}-\frac{\sqrt{14}}{6}-\frac{7\sqrt{2}}{6}-\frac{1}{3}\approx -3.488775824$$

Passi tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{\sqrt{14}+2}{1-\sqrt{7}} billi timmultiplika in-numeratur u d-denominatur mill-1+\sqrt{7}.

$$\frac{\left(\sqrt{14}+2\right)\left(1+\sqrt{7}\right)}{\left(1-\sqrt{7}\right)\left(1+\sqrt{7}\right)}$$

Ikkunsidra li \left(1-\sqrt{7}\right)\left(1+\sqrt{7}\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}.

$$\frac{\left(\sqrt{14}+2\right)\left(1+\sqrt{7}\right)}{1^{2}-\left(\sqrt{7}\right)^{2}}$$

Ikkwadra 1. Ikkwadra \sqrt{7}.

$$\frac{\left(\sqrt{14}+2\right)\left(1+\sqrt{7}\right)}{1-7}$$

Naqqas 7 minn 1 biex tikseb -6.

$$\frac{\left(\sqrt{14}+2\right)\left(1+\sqrt{7}\right)}{-6}$$

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' \sqrt{14}+2 b'kull terminu ta' 1+\sqrt{7}.

$$\frac{\sqrt{14}+\sqrt{14}\sqrt{7}+2+2\sqrt{7}}{-6}$$

Iffattura 14=7\times 2. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{7\times 2} bħala l-prodott tal-għeruq kwadrati \sqrt{7}\sqrt{2}.

$$\frac{\sqrt{14}+\sqrt{7}\sqrt{2}\sqrt{7}+2+2\sqrt{7}}{-6}$$

Immultiplika \sqrt{7} u \sqrt{7} biex tikseb 7.

$$\frac{\sqrt{14}+7\sqrt{2}+2+2\sqrt{7}}{-6}$$

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

$$\frac{-\sqrt{14}-7\sqrt{2}-2-2\sqrt{7}}{6}$$