=frac{sqrt{5}+sqrt{3}}{sqrt{5}-sqrt{3}}+frac{sqrt{5}-sqrt{3}}{sqrt{5}+sqrt{3}} eching

$$= \frac { \sqrt { 5 } + \sqrt { 3 } } { \sqrt { 5 } - \sqrt { 3 } } + \frac { \sqrt { 5 } - \sqrt { 3 } } { \sqrt { 5 } + \sqrt { 3 } }$$

Baholash

$$8$$

Yechim qadamlari

\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} maxrajini \sqrt{5}+\sqrt{3} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$\frac{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

Hisoblang: \left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

\sqrt{5} kvadratini chiqarish. \sqrt{3} kvadratini chiqarish.

$$\frac{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}{5-3}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

2 olish uchun 5 dan 3 ni ayirish.

$$\frac{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}{2}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

\left(\sqrt{5}+\sqrt{3}\right)^{2} hosil qilish uchun \sqrt{5}+\sqrt{3} va \sqrt{5}+\sqrt{3} ni ko'paytirish.

$$\frac{\left(\sqrt{5}+\sqrt{3}\right)^{2}}{2}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(\sqrt{5}+\sqrt{3}\right)^{2} kengaytirilishi uchun ishlating.

$$\frac{\left(\sqrt{5}\right)^{2}+2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{2}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

\sqrt{5} kvadrati – 5.

$$\frac{5+2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{2}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

\sqrt{5} va \sqrt{3} ni koʻpaytirish uchun kvadrat ildiz ichidagi sonlarni koʻpaytiring.

$$\frac{5+2\sqrt{15}+\left(\sqrt{3}\right)^{2}}{2}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

\sqrt{3} kvadrati – 3.

$$\frac{5+2\sqrt{15}+3}{2}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

8 olish uchun 5 va 3'ni qo'shing.

$$\frac{8+2\sqrt{15}}{2}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

4+\sqrt{15} natijani olish uchun 8+2\sqrt{15} ning har bir ifodasini 2 ga bo‘ling.

$$4+\sqrt{15}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}} maxrajini \sqrt{5}-\sqrt{3} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$4+\sqrt{15}+\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}$$

Hisoblang: \left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$4+\sqrt{15}+\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

\sqrt{5} kvadratini chiqarish. \sqrt{3} kvadratini chiqarish.

$$4+\sqrt{15}+\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}{5-3}$$

2 olish uchun 5 dan 3 ni ayirish.

$$4+\sqrt{15}+\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}$$

\left(\sqrt{5}-\sqrt{3}\right)^{2} hosil qilish uchun \sqrt{5}-\sqrt{3} va \sqrt{5}-\sqrt{3} ni ko'paytirish.

$$4+\sqrt{15}+\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(\sqrt{5}-\sqrt{3}\right)^{2} kengaytirilishi uchun ishlating.

$$4+\sqrt{15}+\frac{\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{2}$$

\sqrt{5} kvadrati – 5.

$$4+\sqrt{15}+\frac{5-2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{2}$$

\sqrt{5} va \sqrt{3} ni koʻpaytirish uchun kvadrat ildiz ichidagi sonlarni koʻpaytiring.

$$4+\sqrt{15}+\frac{5-2\sqrt{15}+\left(\sqrt{3}\right)^{2}}{2}$$

\sqrt{3} kvadrati – 3.

$$4+\sqrt{15}+\frac{5-2\sqrt{15}+3}{2}$$

8 olish uchun 5 va 3'ni qo'shing.

$$4+\sqrt{15}+\frac{8-2\sqrt{15}}{2}$$

4-\sqrt{15} natijani olish uchun 8-2\sqrt{15} ning har bir ifodasini 2 ga bo‘ling.

$$4+\sqrt{15}+4-\sqrt{15}$$

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

$$8+\sqrt{15}-\sqrt{15}$$

0 ni olish uchun \sqrt{15} va -\sqrt{15} ni birlashtirish.

$$8$$

Omil

$$2^{3}$$