Issolvi (sqrt{6}-sqrt{2})^2-frac{sqrt{6}-sqrt{2}}{sqrt{6}+sqrt{2}}

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

Evalwa

$$6-3\sqrt{3}\approx 0.803847577$$

Passi tas-Soluzzjonijiet

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(\sqrt{6}-\sqrt{2}\right)^{2}.

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

Il-kwadrat ta' \sqrt{6} huwa 6.

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

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

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

Immultiplika \sqrt{2} u \sqrt{2} biex tikseb 2.

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

Immultiplika -2 u 2 biex tikseb -4.

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

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

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

Żid 6 u 2 biex tikseb 8.

$$8-4\sqrt{3}-\frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}+\sqrt{2}}$$

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

$$8-4\sqrt{3}-\frac{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right)}{\left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right)}$$

Ikkunsidra li \left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\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}.

$$8-4\sqrt{3}-\frac{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Ikkwadra \sqrt{6}. Ikkwadra \sqrt{2}.

$$8-4\sqrt{3}-\frac{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right)}{6-2}$$

Naqqas 2 minn 6 biex tikseb 4.

$$8-4\sqrt{3}-\frac{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right)}{4}$$

Immultiplika \sqrt{6}-\sqrt{2} u \sqrt{6}-\sqrt{2} biex tikseb \left(\sqrt{6}-\sqrt{2}\right)^{2}.

$$8-4\sqrt{3}-\frac{\left(\sqrt{6}-\sqrt{2}\right)^{2}}{4}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(\sqrt{6}-\sqrt{2}\right)^{2}.

$$8-4\sqrt{3}-\frac{\left(\sqrt{6}\right)^{2}-2\sqrt{6}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{4}$$

Il-kwadrat ta' \sqrt{6} huwa 6.

$$8-4\sqrt{3}-\frac{6-2\sqrt{6}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{4}$$

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

$$8-4\sqrt{3}-\frac{6-2\sqrt{2}\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{4}$$

Immultiplika \sqrt{2} u \sqrt{2} biex tikseb 2.

$$8-4\sqrt{3}-\frac{6-2\times 2\sqrt{3}+\left(\sqrt{2}\right)^{2}}{4}$$

Immultiplika -2 u 2 biex tikseb -4.

$$8-4\sqrt{3}-\frac{6-4\sqrt{3}+\left(\sqrt{2}\right)^{2}}{4}$$

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

$$8-4\sqrt{3}-\frac{6-4\sqrt{3}+2}{4}$$

Żid 6 u 2 biex tikseb 8.

$$8-4\sqrt{3}-\frac{8-4\sqrt{3}}{4}$$

Iddividi kull terminu ta' 8-4\sqrt{3} b'4 biex tikseb2-\sqrt{3}.

$$8-4\sqrt{3}-\left(2-\sqrt{3}\right)$$

Biex issib l-oppost ta' 2-\sqrt{3}, sib l-oppost ta' kull terminu.

$$8-4\sqrt{3}-2+\sqrt{3}$$

Naqqas 2 minn 8 biex tikseb 6.

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

Ikkombina -4\sqrt{3} u \sqrt{3} biex tikseb -3\sqrt{3}.

$$6-3\sqrt{3}$$

Fattur

$$3 {(2 - \sqrt{3})} = 0.803847577$$