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

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

Arvuta

$$2\sqrt{3}\approx 3,464101615$$

Lahenduse etapid

Kasutage distributiivsusomadust, et korrutada 1+\sqrt{2}+\sqrt{3} ja 2+\sqrt{2}-\sqrt{6}, ning koondage sarnased liikmed.

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

\sqrt{2} ruut on 2.

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

Liitke 2 ja 2, et leida 4.

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

Tegurda 6=2\times 3. Kirjutage \sqrt{2\times 3} toote juured, kui see ruut \sqrt{2}\sqrt{3}.

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

Korrutage \sqrt{2} ja \sqrt{2}, et leida 2.

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

Kombineerige -2\sqrt{3} ja 2\sqrt{3}, et leida 0.

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

\sqrt{3} ja \sqrt{2} korrutage numbrid, mis on sama juur.

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

Kombineerige -\sqrt{6} ja \sqrt{6}, et leida 0.

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

Tegurda 6=3\times 2. Kirjutage \sqrt{3\times 2} toote juured, kui see ruut \sqrt{3}\sqrt{2}.

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

Korrutage \sqrt{3} ja \sqrt{3}, et leida 3.

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

Kombineerige 3\sqrt{2} ja -3\sqrt{2}, et leida 0.

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

Kasutage kaksliikme \left(\sqrt{3}-1\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

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

\sqrt{3} ruut on 3.

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

Liitke 3 ja 1, et leida 4.

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

Avaldise "4-2\sqrt{3}" vastandi leidmiseks tuleb leida iga liikme vastand.

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

Lahutage 4 väärtusest 4, et leida 0.

$$2\sqrt{3}$$