frac{sqrt{7}+sqrt{5}}{sqrt{7}-sqrt{5}}+frac{sqrt{7}-sqrt{5}}{sqrt{7}+sqrt{5}}= eching

$$\frac { \sqrt { 7 } + \sqrt { 5 } } { \sqrt { 7 } - \sqrt { 5 } } + \frac { \sqrt { 7 } - \sqrt { 5 } } { \sqrt { 7 } + \sqrt { 5 } } =$$

Baholash

$$12$$

Yechim qadamlari

\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}} maxrajini \sqrt{7}+\sqrt{5} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Hisoblang: \left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{5}\right)^{2}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

\sqrt{7} kvadratini chiqarish. \sqrt{5} kvadratini chiqarish.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{7-5}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

2 olish uchun 7 dan 5 ni ayirish.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

\left(\sqrt{7}+\sqrt{5}\right)^{2} hosil qilish uchun \sqrt{7}+\sqrt{5} va \sqrt{7}+\sqrt{5} ni ko'paytirish.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)^{2}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(\sqrt{7}+\sqrt{5}\right)^{2} kengaytirilishi uchun ishlating.

$$\frac{\left(\sqrt{7}\right)^{2}+2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

\sqrt{7} kvadrati – 7.

$$\frac{7+2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

\sqrt{7} va \sqrt{5} ni koʻpaytirish uchun kvadrat ildiz ichidagi sonlarni koʻpaytiring.

$$\frac{7+2\sqrt{35}+\left(\sqrt{5}\right)^{2}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

\sqrt{5} kvadrati – 5.

$$\frac{7+2\sqrt{35}+5}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

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

$$\frac{12+2\sqrt{35}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

6+\sqrt{35} natijani olish uchun 12+2\sqrt{35} ning har bir ifodasini 2 ga bo‘ling.

$$6+\sqrt{35}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}} maxrajini \sqrt{7}-\sqrt{5} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}$$

Hisoblang: \left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

\sqrt{7} kvadratini chiqarish. \sqrt{5} kvadratini chiqarish.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{7-5}$$

2 olish uchun 7 dan 5 ni ayirish.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{2}$$

\left(\sqrt{7}-\sqrt{5}\right)^{2} hosil qilish uchun \sqrt{7}-\sqrt{5} va \sqrt{7}-\sqrt{5} ni ko'paytirish.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)^{2}}{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(\sqrt{7}-\sqrt{5}\right)^{2} kengaytirilishi uchun ishlating.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}\right)^{2}-2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2}$$

\sqrt{7} kvadrati – 7.

$$6+\sqrt{35}+\frac{7-2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2}$$

\sqrt{7} va \sqrt{5} ni koʻpaytirish uchun kvadrat ildiz ichidagi sonlarni koʻpaytiring.

$$6+\sqrt{35}+\frac{7-2\sqrt{35}+\left(\sqrt{5}\right)^{2}}{2}$$

\sqrt{5} kvadrati – 5.

$$6+\sqrt{35}+\frac{7-2\sqrt{35}+5}{2}$$

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

$$6+\sqrt{35}+\frac{12-2\sqrt{35}}{2}$$

6-\sqrt{35} natijani olish uchun 12-2\sqrt{35} ning har bir ifodasini 2 ga bo‘ling.

$$6+\sqrt{35}+6-\sqrt{35}$$

12 olish uchun 6 va 6'ni qo'shing.

$$12+\sqrt{35}-\sqrt{35}$$

0 ni olish uchun \sqrt{35} va -\sqrt{35} ni birlashtirish.

$$12$$

Omil

$$2^{2}\times 3$$