Issolvi frac{1/sqrt{2}-1}{1/sqrt{2}+sqrt{3}}

$$\frac{ \frac{ 1 }{ \sqrt{ 2 } } -1 }{ \frac{ 1 }{ \sqrt{ 2 } } + \sqrt{ 3 } }$$

Evalwa

$$\frac{\sqrt{2}+\sqrt{6}-2\sqrt{3}-1}{5}\approx -0.120079662$$

Passi Qosra tas-Soluzzjonijiet

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

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

Il-kwadrat ta' \sqrt{2} huwa 2.

$$\frac{\frac{\sqrt{2}}{2}-1}{\frac{1}{\sqrt{2}}+\sqrt{3}}$$

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. Immultiplika 1 b'\frac{2}{2}.

$$\frac{\frac{\sqrt{2}}{2}-\frac{2}{2}}{\frac{1}{\sqrt{2}}+\sqrt{3}}$$

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

$$\frac{\frac{\sqrt{2}-2}{2}}{\frac{1}{\sqrt{2}}+\sqrt{3}}$$

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

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

Il-kwadrat ta' \sqrt{2} huwa 2.

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

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. Immultiplika \sqrt{3} b'\frac{2}{2}.

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

Billi \frac{\sqrt{2}}{2} u \frac{2\sqrt{3}}{2} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

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

Iddividi \frac{\sqrt{2}-2}{2} b'\frac{\sqrt{2}+2\sqrt{3}}{2} billi timmultiplika \frac{\sqrt{2}-2}{2} bir-reċiproku ta' \frac{\sqrt{2}+2\sqrt{3}}{2}.

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

Annulla 2 fin-numeratur u d-denominatur.

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

Irrazzjonalizza d-denominatur tal-\frac{\sqrt{2}-2}{\sqrt{2}+2\sqrt{3}} billi timmultiplika in-numeratur u d-denominatur mill-\sqrt{2}-2\sqrt{3}.

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

Ikkunsidra li \left(\sqrt{2}+2\sqrt{3}\right)\left(\sqrt{2}-2\sqrt{3}\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}.

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

Il-kwadrat ta' \sqrt{2} huwa 2.

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

Espandi \left(2\sqrt{3}\right)^{2}.

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

Ikkalkula 2 bil-power ta' 2 u tikseb 4.

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

Il-kwadrat ta' \sqrt{3} huwa 3.

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

Immultiplika 4 u 3 biex tikseb 12.

$$\frac{\left(\sqrt{2}-2\right)\left(\sqrt{2}-2\sqrt{3}\right)}{2-12}$$

Naqqas 12 minn 2 biex tikseb -10.

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

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' \sqrt{2}-2 b'kull terminu ta' \sqrt{2}-2\sqrt{3}.

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

Il-kwadrat ta' \sqrt{2} huwa 2.

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

Biex timmultiplika \sqrt{2} u \sqrt{3}, immultiplika n-numri taħt l-għerq kwadrat.

$$\frac{2-2\sqrt{6}-2\sqrt{2}+4\sqrt{3}}{-10}$$