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

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

Baholash

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

Yechim qadamlari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(\sqrt{2}+\sqrt{5}\right)^{2} kengaytirilishi uchun ishlating.

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

\sqrt{2} kvadrati – 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} va \sqrt{5} ni koʻpaytirish uchun kvadrat ildiz ichidagi sonlarni koʻpaytiring.

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

\sqrt{5} kvadrati – 5.

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

7 olish uchun 2 va 5'ni qo'shing.

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2+\sqrt{10}\right)^{2} kengaytirilishi uchun ishlating.

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

\sqrt{10} kvadrati – 10.

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

14 olish uchun 4 va 10'ni qo'shing.

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

14+4\sqrt{10} teskarisini topish uchun har birining teskarisini toping.

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

-7 olish uchun 7 dan 14 ni ayirish.

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

-2\sqrt{10} ni olish uchun 2\sqrt{10} va -4\sqrt{10} ni birlashtirish.

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

Faktor: 90=3^{2}\times 10. \sqrt{3^{2}\times 10} koʻpaytmasining kvadrat ildizini \sqrt{3^{2}}\sqrt{10} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 3^{2} ning kvadrat ildizini chiqarish.

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

\sqrt{10} ni olish uchun -2\sqrt{10} va 3\sqrt{10} ni birlashtirish.

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

Hisoblang: \left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 1 kvadratini chiqarish.

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

\left(2\sqrt{2}\right)^{2} ni kengaytirish.

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

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

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

\sqrt{2} kvadrati – 2.

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

8 hosil qilish uchun 4 va 2 ni ko'paytirish.

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

7 olish uchun 8 dan 1 ni ayirish.

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

0 olish uchun -7 va 7'ni qo'shing.

$$\sqrt{10}$$