Réitigh frac{sqrt{12}+sqrt{6}+sqrt{2}+2}{sqrt{3}+1}

$$\frac { \sqrt { 12 } + \sqrt { 6 } + \sqrt { 2 } + 2 } { \sqrt { 3 } + 1 }$$

Luacháil

$$\sqrt{2}+2\approx 3.414213562$$

Céimeanna Réitigh

Fachtóirigh 12=2^{2}\times 3. Athscríobh fréamh cearnach an toraidh \sqrt{2^{2}\times 3} mar thoradh na bhfréamhacha cearnacha \sqrt{2^{2}}\sqrt{3}. Tóg fréamh chearnach 2^{2}.

$$\frac{2\sqrt{3}+\sqrt{6}+\sqrt{2}+2}{\sqrt{3}+1}$$

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi \sqrt{3}-1 chun ainmneoir \frac{2\sqrt{3}+\sqrt{6}+\sqrt{2}+2}{\sqrt{3}+1} a thiontú in uimhir chóimheasta.

$$\frac{\left(2\sqrt{3}+\sqrt{6}+\sqrt{2}+2\right)\left(\sqrt{3}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}$$

Mar shampla \left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right). Is féidir iolrúchán a athrú ó bhonn go dtí difríocht na gcearnóg ag úsáid na rialach seo: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2\sqrt{3}+\sqrt{6}+\sqrt{2}+2\right)\left(\sqrt{3}-1\right)}{\left(\sqrt{3}\right)^{2}-1^{2}}$$

Cearnóg \sqrt{3}. Cearnóg 1.

$$\frac{\left(2\sqrt{3}+\sqrt{6}+\sqrt{2}+2\right)\left(\sqrt{3}-1\right)}{3-1}$$

Dealaigh 1 ó 3 chun 2 a fháil.

$$\frac{\left(2\sqrt{3}+\sqrt{6}+\sqrt{2}+2\right)\left(\sqrt{3}-1\right)}{2}$$

Cuir an t-airí dáileacháin i bhfeidhm trí gach téarma de 2\sqrt{3}+\sqrt{6}+\sqrt{2}+2 a iolrú faoi gach téarma de \sqrt{3}-1.

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

Is é 3 uimhir chearnach \sqrt{3}.

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

Méadaigh 2 agus 3 chun 6 a fháil.

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

Fachtóirigh 6=3\times 2. Athscríobh fréamh cearnach an toraidh \sqrt{3\times 2} mar thoradh na bhfréamhacha cearnacha \sqrt{3}\sqrt{2}.

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

Méadaigh \sqrt{3} agus \sqrt{3} chun 3 a fháil.

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

Iolraigh na huimhreacha faoin bhfréamh cearnach chun \sqrt{2} agus \sqrt{3} a iolrú.

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

Comhcheangail -\sqrt{6} agus \sqrt{6} chun 0 a fháil.

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

Comhcheangail 3\sqrt{2} agus -\sqrt{2} chun 2\sqrt{2} a fháil.

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

Comhcheangail -2\sqrt{3} agus 2\sqrt{3} chun 0 a fháil.

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

Dealaigh 2 ó 6 chun 4 a fháil.

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

Roinn 4+2\sqrt{2} faoi 2 chun 2+\sqrt{2} a fháil.

$$2+\sqrt{2}$$