Rešite frac{sqrt{2}}{sqrt{6}-sqrt{2}}-frac{sqrt{3}}{sqrt{6}+sqrt{2}}

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

Ovrednoti

$$\frac{\sqrt{6}+2\sqrt{3}+2-3\sqrt{2}}{4}\approx 0,917737668$$

Kratka navodila za rešitev

Racionalizirajte imenovalec \frac{\sqrt{2}}{\sqrt{6}-\sqrt{2}} tako, da pomnožite števec in imenovalec s \sqrt{6}+\sqrt{2}.

$$\frac{\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}+\sqrt{2}\right)}-\frac{\sqrt{3}}{\sqrt{6}+\sqrt{2}}$$

Razmislite o \left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}+\sqrt{2}\right). Množenje je lahko preoblikovano v razliko kvadratov s pravilom: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{2}\right)^{2}}-\frac{\sqrt{3}}{\sqrt{6}+\sqrt{2}}$$

Kvadrat števila \sqrt{6}. Kvadrat števila \sqrt{2}.

$$\frac{\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)}{6-2}-\frac{\sqrt{3}}{\sqrt{6}+\sqrt{2}}$$

Odštejte 2 od 6, da dobite 4.

$$\frac{\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)}{4}-\frac{\sqrt{3}}{\sqrt{6}+\sqrt{2}}$$

Racionalizirajte imenovalec \frac{\sqrt{3}}{\sqrt{6}+\sqrt{2}} tako, da pomnožite števec in imenovalec s \sqrt{6}-\sqrt{2}.

$$\frac{\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)}{4}-\frac{\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{\left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right)}$$

Razmislite o \left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right). Množenje je lahko preoblikovano v razliko kvadratov s pravilom: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)}{4}-\frac{\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Kvadrat števila \sqrt{6}. Kvadrat števila \sqrt{2}.

$$\frac{\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)}{4}-\frac{\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{6-2}$$

Odštejte 2 od 6, da dobite 4.

$$\frac{\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)}{4}-\frac{\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4}$$

Ker \frac{\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)}{4} in \frac{\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4} imata isti imenovalec, jih odštejte tako, da odštejete njihove števce.

$$\frac{\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)-\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4}$$

Izvedi množenje v \sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)-\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right).

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