Issolvi (1+sqrt{2}+sqrt{3})(2+sqrt{2}-sqrt{6})-(sqrt{3}-1)^2

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

Evalwa

$$2\sqrt{3}\approx 3.464101615$$

Passi tas-Soluzzjonijiet

Uża l-propjetà distributtiva biex timmultiplika 1+\sqrt{2}+\sqrt{3} b'2+\sqrt{2}-\sqrt{6} u kkombina termini simili.

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

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

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

Żid 2 u 2 biex tikseb 4.

$$4+3\sqrt{2}-\sqrt{6}-\sqrt{2}\sqrt{6}+2\sqrt{3}+\sqrt{3}\sqrt{2}-\sqrt{3}\sqrt{6}-\left(\sqrt{3}-1\right)^{2}$$

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}.

$$4+3\sqrt{2}-\sqrt{6}-\sqrt{2}\sqrt{2}\sqrt{3}+2\sqrt{3}+\sqrt{3}\sqrt{2}-\sqrt{3}\sqrt{6}-\left(\sqrt{3}-1\right)^{2}$$

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

$$4+3\sqrt{2}-\sqrt{6}-2\sqrt{3}+2\sqrt{3}+\sqrt{3}\sqrt{2}-\sqrt{3}\sqrt{6}-\left(\sqrt{3}-1\right)^{2}$$

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

$$4+3\sqrt{2}-\sqrt{6}+\sqrt{3}\sqrt{2}-\sqrt{3}\sqrt{6}-\left(\sqrt{3}-1\right)^{2}$$

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

$$4+3\sqrt{2}-\sqrt{6}+\sqrt{6}-\sqrt{3}\sqrt{6}-\left(\sqrt{3}-1\right)^{2}$$

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

$$4+3\sqrt{2}-\sqrt{3}\sqrt{6}-\left(\sqrt{3}-1\right)^{2}$$

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}.

$$4+3\sqrt{2}-\sqrt{3}\sqrt{3}\sqrt{2}-\left(\sqrt{3}-1\right)^{2}$$

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

$$4+3\sqrt{2}-3\sqrt{2}-\left(\sqrt{3}-1\right)^{2}$$

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

$$4-\left(\sqrt{3}-1\right)^{2}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(\sqrt{3}-1\right)^{2}.

$$4-\left(\left(\sqrt{3}\right)^{2}-2\sqrt{3}+1\right)$$

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

$$4-\left(3-2\sqrt{3}+1\right)$$

Żid 3 u 1 biex tikseb 4.

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

Biex issib l-oppost ta' 4-2\sqrt{3}, sib l-oppost ta' kull terminu.

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

Naqqas 4 minn 4 biex tikseb 0.

$$2\sqrt{3}$$