Lahenda (sqrt{18}+sqrt{12})(3sqrt{2}-2sqrt{3})-(sqrt{3}-sqrt{2})^2

$$( \sqrt { 18 } + \sqrt { 12 } ) ( 3 \sqrt { 2 } - 2 \sqrt { 3 } ) - ( \sqrt { 3 } - \sqrt { 2 } ) ^ { 2 }$$

Arvuta

$$2\sqrt{6}+1\approx 5,898979486$$

Lahenduse etapid

Tegurda 18=3^{2}\times 2. Kirjutage \sqrt{3^{2}\times 2} toote juured, kui see ruut \sqrt{3^{2}}\sqrt{2}. Leidke 3^{2} ruutjuur.

$$\left(3\sqrt{2}+\sqrt{12}\right)\left(3\sqrt{2}-2\sqrt{3}\right)-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Tegurda 12=2^{2}\times 3. Kirjutage \sqrt{2^{2}\times 3} toote juured, kui see ruut \sqrt{2^{2}}\sqrt{3}. Leidke 2^{2} ruutjuur.

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

Mõelge valemile \left(3\sqrt{2}+2\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Laiendage \left(3\sqrt{2}\right)^{2}.

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

Arvutage 2 aste 3 ja leidke 9.

$$9\left(\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

\sqrt{2} ruut on 2.

$$9\times 2-\left(2\sqrt{3}\right)^{2}-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Korrutage 9 ja 2, et leida 18.

$$18-\left(2\sqrt{3}\right)^{2}-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Laiendage \left(2\sqrt{3}\right)^{2}.

$$18-2^{2}\left(\sqrt{3}\right)^{2}-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Arvutage 2 aste 2 ja leidke 4.

$$18-4\left(\sqrt{3}\right)^{2}-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

\sqrt{3} ruut on 3.

$$18-4\times 3-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Korrutage 4 ja 3, et leida 12.

$$18-12-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Lahutage 12 väärtusest 18, et leida 6.

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

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

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

\sqrt{3} ruut on 3.

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

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

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

\sqrt{2} ruut on 2.

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

Liitke 3 ja 2, et leida 5.

$$6-\left(5-2\sqrt{6}\right)$$

Avaldise "5-2\sqrt{6}" vastandi leidmiseks tuleb leida iga liikme vastand.

$$6-5+2\sqrt{6}$$

Lahutage 5 väärtusest 6, et leida 1.

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