Issolvi {left(sqrt{6}+sqrt{2}right)}^2-2sqrt{2}*sqrt{6}+sqrt{2}

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

Evalwa

$$\sqrt{2}+8\approx 9.414213562$$

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}.

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

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

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

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}.

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

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

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

Immultiplika 2 u 2 biex tikseb 4.

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

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

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

Żid 6 u 2 biex tikseb 8.

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

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}.

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

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

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

Immultiplika 2 u 2 biex tikseb 4.

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

Ikkombina 4\sqrt{3} u -4\sqrt{3} biex tikseb 0.

$$8+\sqrt{2}$$