frac{8+4-2sqrt{5}-4sqrt{5}+2sqrt{10}}{1-sqrt{5}} eching

$$\frac { 8 + 4 - 2 \sqrt { 5 } - 4 \sqrt { 5 } + 2 \sqrt { 10 } } { 1 - \sqrt { 5 } }$$

Baholash

$$\frac{9-\sqrt{10}-3\sqrt{5}-5\sqrt{2}}{2}\approx -3,970774702$$

Yechim qadamlari

12 olish uchun 8 va 4'ni qo'shing.

$$\frac{12-2\sqrt{5}-4\sqrt{5}+2\sqrt{10}}{1-\sqrt{5}}$$

-6\sqrt{5} ni olish uchun -2\sqrt{5} va -4\sqrt{5} ni birlashtirish.

$$\frac{12-6\sqrt{5}+2\sqrt{10}}{1-\sqrt{5}}$$

\frac{12-6\sqrt{5}+2\sqrt{10}}{1-\sqrt{5}} maxrajini 1+\sqrt{5} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)}$$

Hisoblang: \left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{1^{2}-\left(\sqrt{5}\right)^{2}}$$

1 kvadratini chiqarish. \sqrt{5} kvadratini chiqarish.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{1-5}$$

-4 olish uchun 1 dan 5 ni ayirish.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{-4}$$

12-6\sqrt{5}+2\sqrt{10} ifodaning har bir elementini 1+\sqrt{5} ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$\frac{12+12\sqrt{5}-6\sqrt{5}-6\left(\sqrt{5}\right)^{2}+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

6\sqrt{5} ni olish uchun 12\sqrt{5} va -6\sqrt{5} ni birlashtirish.

$$\frac{12+6\sqrt{5}-6\left(\sqrt{5}\right)^{2}+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

\sqrt{5} kvadrati – 5.

$$\frac{12+6\sqrt{5}-6\times 5+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

-30 hosil qilish uchun -6 va 5 ni ko'paytirish.

$$\frac{12+6\sqrt{5}-30+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

-18 olish uchun 12 dan 30 ni ayirish.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Faktor: 10=5\times 2. \sqrt{5\times 2} koʻpaytmasining kvadrat ildizini \sqrt{5}\sqrt{2} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+2\sqrt{5}\sqrt{2}\sqrt{5}}{-4}$$

5 hosil qilish uchun \sqrt{5} va \sqrt{5} ni ko'paytirish.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+2\times 5\sqrt{2}}{-4}$$

10 hosil qilish uchun 2 va 5 ni ko'paytirish.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+10\sqrt{2}}{-4}$$

Omil

$$\frac{9 - \sqrt{10} - 3 \sqrt{5} - 5 \sqrt{2}}{2} = -3,9707747022666124$$