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

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

Izrēķināt

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

Atrisinājuma darbības

Lietojiet Ņūtona binomu \left(a+b\right)^{2}=a^{2}+2ab+b^{2}, lai izvērstu \left(\sqrt{2}+\sqrt{5}\right)^{2}.

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

Skaitļa \sqrt{2} kvadrāts ir 2.

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

Lai reiziniet \sqrt{2} un \sqrt{5}, reiziniet numurus zem kvadrātveida saknes.

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

Skaitļa \sqrt{5} kvadrāts ir 5.

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

Saskaitiet 2 un 5, lai iegūtu 7.

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

Lietojiet Ņūtona binomu \left(a+b\right)^{2}=a^{2}+2ab+b^{2}, lai izvērstu \left(2+\sqrt{10}\right)^{2}.

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

Skaitļa \sqrt{10} kvadrāts ir 10.

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

Saskaitiet 4 un 10, lai iegūtu 14.

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

Lai atrastu 14+4\sqrt{10} pretējo vērtību, atrodiet katra locekļa pretējo vērtību.

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

Atņemiet 14 no 7, lai iegūtu -7.

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

Savelciet 2\sqrt{10} un -4\sqrt{10}, lai iegūtu -2\sqrt{10}.

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

Sadaliet reizinātājos 90=3^{2}\times 10. Pārrakstiet reizinājuma kvadrātsakni \sqrt{3^{2}\times 10} kā kvadrātveida saknes \sqrt{3^{2}}\sqrt{10}. Izvelciet kvadrātsakni no 3^{2}.

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

Savelciet -2\sqrt{10} un 3\sqrt{10}, lai iegūtu \sqrt{10}.

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

Apsveriet \left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right). Reizināšanu var pārvērst par kvadrātu starpību, izmantojot šo kārtulu: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Kāpiniet 1 kvadrātā.

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

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

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

Aprēķiniet 2 pakāpē 2 un iegūstiet 4.

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

Skaitļa \sqrt{2} kvadrāts ir 2.

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

Reiziniet 4 un 2, lai iegūtu 8.

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

Atņemiet 1 no 8, lai iegūtu 7.

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

Saskaitiet -7 un 7, lai iegūtu 0.

$$\sqrt{10}$$