Lahenda (sqrt{2}+sqrt{5})^2-(2+sqrt{10})^2+sqrt{90}+(2sqrt{2}-1)(2sqrt{2}+1)

$$( \sqrt { 2 } + \sqrt { 5 } ) ^ { 2 } - ( 2 + \sqrt { 10 } ) ^ { 2 } + \sqrt { 90 } + ( 2 \sqrt { 2 } - 1 ) ( 2 \sqrt { 2 } + 1 )$$

Arvuta

$$\sqrt{10}\approx 3,16227766$$

Lahenduse etapid

Kasutage kaksliikme \left(\sqrt{2}+\sqrt{5}\right)^{2} arendamiseks binoomvalemit \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

$$\left(\sqrt{2}\right)^{2}+2\sqrt{2}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

\sqrt{2} ruut on 2.

$$2+2\sqrt{2}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

\sqrt{2} ja \sqrt{5} korrutage numbrid, mis on sama juur.

$$2+2\sqrt{10}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

\sqrt{5} ruut on 5.

$$2+2\sqrt{10}+5-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Liitke 2 ja 5, et leida 7.

$$7+2\sqrt{10}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Kasutage kaksliikme \left(2+\sqrt{10}\right)^{2} arendamiseks binoomvalemit \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

$$7+2\sqrt{10}-\left(4+4\sqrt{10}+\left(\sqrt{10}\right)^{2}\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

\sqrt{10} ruut on 10.

$$7+2\sqrt{10}-\left(4+4\sqrt{10}+10\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Liitke 4 ja 10, et leida 14.

$$7+2\sqrt{10}-\left(14+4\sqrt{10}\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Avaldise "14+4\sqrt{10}" vastandi leidmiseks tuleb leida iga liikme vastand.

$$7+2\sqrt{10}-14-4\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Lahutage 14 väärtusest 7, et leida -7.

$$-7+2\sqrt{10}-4\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Kombineerige 2\sqrt{10} ja -4\sqrt{10}, et leida -2\sqrt{10}.

$$-7-2\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Tegurda 90=3^{2}\times 10. Kirjutage \sqrt{3^{2}\times 10} toote juured, kui see ruut \sqrt{3^{2}}\sqrt{10}. Leidke 3^{2} ruutjuur.

$$-7-2\sqrt{10}+3\sqrt{10}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Kombineerige -2\sqrt{10} ja 3\sqrt{10}, et leida \sqrt{10}.

$$-7+\sqrt{10}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Mõelge valemile \left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Tõstke 1 ruutu.

$$-7+\sqrt{10}+\left(2\sqrt{2}\right)^{2}-1$$

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

$$-7+\sqrt{10}+2^{2}\left(\sqrt{2}\right)^{2}-1$$

Arvutage 2 aste 2 ja leidke 4.

$$-7+\sqrt{10}+4\left(\sqrt{2}\right)^{2}-1$$

\sqrt{2} ruut on 2.

$$-7+\sqrt{10}+4\times 2-1$$

Korrutage 4 ja 2, et leida 8.

$$-7+\sqrt{10}+8-1$$

Lahutage 1 väärtusest 8, et leida 7.

$$-7+\sqrt{10}+7$$

Liitke -7 ja 7, et leida 0.

$$\sqrt{10}$$