frac{4-sqrt{2}}{4+sqrt{2}}-frac{4+sqrt{2}}{4-sqrt{2}} eching

$$\frac { 4 - \sqrt { 2 } } { 4 + \sqrt { 2 } } - \frac { 4 + \sqrt { 2 } } { 4 - \sqrt { 2 } }$$

Baholash

$$-\frac{8\sqrt{2}}{7}\approx -1,616244071$$

Qisqa yechim qadamlari

\frac{4-\sqrt{2}}{4+\sqrt{2}} maxrajini 4-\sqrt{2} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$\frac{\left(4-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{\left(4+\sqrt{2}\right)\left(4-\sqrt{2}\right)}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Hisoblang: \left(4+\sqrt{2}\right)\left(4-\sqrt{2}\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(4-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{4^{2}-\left(\sqrt{2}\right)^{2}}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

4 kvadratini chiqarish. \sqrt{2} kvadratini chiqarish.

$$\frac{\left(4-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{16-2}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

14 olish uchun 16 dan 2 ni ayirish.

$$\frac{\left(4-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

\left(4-\sqrt{2}\right)^{2} hosil qilish uchun 4-\sqrt{2} va 4-\sqrt{2} ni ko'paytirish.

$$\frac{\left(4-\sqrt{2}\right)^{2}}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(4-\sqrt{2}\right)^{2} kengaytirilishi uchun ishlating.

$$\frac{16-8\sqrt{2}+\left(\sqrt{2}\right)^{2}}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

\sqrt{2} kvadrati – 2.

$$\frac{16-8\sqrt{2}+2}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

18 olish uchun 16 va 2'ni qo'shing.

$$\frac{18-8\sqrt{2}}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

\frac{4+\sqrt{2}}{4-\sqrt{2}} maxrajini 4+\sqrt{2} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{\left(4-\sqrt{2}\right)\left(4+\sqrt{2}\right)}$$

Hisoblang: \left(4-\sqrt{2}\right)\left(4+\sqrt{2}\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{4^{2}-\left(\sqrt{2}\right)^{2}}$$

4 kvadratini chiqarish. \sqrt{2} kvadratini chiqarish.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{16-2}$$

14 olish uchun 16 dan 2 ni ayirish.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{14}$$

\left(4+\sqrt{2}\right)^{2} hosil qilish uchun 4+\sqrt{2} va 4+\sqrt{2} ni ko'paytirish.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)^{2}}{14}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(4+\sqrt{2}\right)^{2} kengaytirilishi uchun ishlating.

$$\frac{18-8\sqrt{2}}{14}-\frac{16+8\sqrt{2}+\left(\sqrt{2}\right)^{2}}{14}$$

\sqrt{2} kvadrati – 2.

$$\frac{18-8\sqrt{2}}{14}-\frac{16+8\sqrt{2}+2}{14}$$

18 olish uchun 16 va 2'ni qo'shing.

$$\frac{18-8\sqrt{2}}{14}-\frac{18+8\sqrt{2}}{14}$$

\frac{18-8\sqrt{2}}{14} va \frac{18+8\sqrt{2}}{14} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{18-8\sqrt{2}-\left(18+8\sqrt{2}\right)}{14}$$

18-8\sqrt{2}-\left(18+8\sqrt{2}\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{18-8\sqrt{2}-18-8\sqrt{2}}{14}$$

18-8\sqrt{2}-18-8\sqrt{2} hisob-kitobini qiling.

$$\frac{-16\sqrt{2}}{14}$$

-\frac{8}{7}\sqrt{2} ni olish uchun -16\sqrt{2} ni 14 ga bo‘ling.

$$-\frac{8}{7}\sqrt{2}$$