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

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

Evalwa

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

Passi tas-Soluzzjonijiet

Iffattura 12=2^{2}\times 3. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{2^{2}\times 3} bħala l-prodott tal-għeruq kwadrati \sqrt{2^{2}}\sqrt{3}. Ħu l-għerq kwadrat ta' 2^{2}.

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

Irrazzjonalizza d-denominatur tal-\frac{2\sqrt{3}+\sqrt{6}+\sqrt{2}+2}{\sqrt{3}+1} billi timmultiplika in-numeratur u d-denominatur mill-\sqrt{3}-1.

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

Ikkunsidra li \left(\sqrt{3}+1\right)\left(\sqrt{3}-1\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}.

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

Ikkwadra \sqrt{3}. Ikkwadra 1.

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

Naqqas 1 minn 3 biex tikseb 2.

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

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 2\sqrt{3}+\sqrt{6}+\sqrt{2}+2 b'kull terminu ta' \sqrt{3}-1.

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

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

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

Immultiplika 2 u 3 biex tikseb 6.

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

Iffattura 6=3\times 2. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{3\times 2} bħala l-prodott tal-għeruq kwadrati \sqrt{3}\sqrt{2}.

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

Immultiplika \sqrt{3} u \sqrt{3} biex tikseb 3.

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

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

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

Ikkombina -\sqrt{6} u \sqrt{6} biex tikseb 0.

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

Ikkombina 3\sqrt{2} u -\sqrt{2} biex tikseb 2\sqrt{2}.

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

Ikkombina -2\sqrt{3} u 2\sqrt{3} biex tikseb 0.

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

Naqqas 2 minn 6 biex tikseb 4.

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

Iddividi kull terminu ta' 4+2\sqrt{2} b'2 biex tikseb2+\sqrt{2}.

$$2+\sqrt{2}$$