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

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

Baholash

$$\frac{32}{9}\approx 3,555555556$$

Qisqa yechim qadamlari

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

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

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

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

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

$$\frac{\left(5-\sqrt{7}\right)\left(5-\sqrt{7}\right)}{25-7}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

18 olish uchun 25 dan 7 ni ayirish.

$$\frac{\left(5-\sqrt{7}\right)\left(5-\sqrt{7}\right)}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

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

$$\frac{\left(5-\sqrt{7}\right)^{2}}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

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

$$\frac{25-10\sqrt{7}+\left(\sqrt{7}\right)^{2}}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

\sqrt{7} kvadrati – 7.

$$\frac{25-10\sqrt{7}+7}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

32 olish uchun 25 va 7'ni qo'shing.

$$\frac{32-10\sqrt{7}}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

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

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)\left(5+\sqrt{7}\right)}{\left(5-\sqrt{7}\right)\left(5+\sqrt{7}\right)}$$

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

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)\left(5+\sqrt{7}\right)}{5^{2}-\left(\sqrt{7}\right)^{2}}$$

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

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)\left(5+\sqrt{7}\right)}{25-7}$$

18 olish uchun 25 dan 7 ni ayirish.

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)\left(5+\sqrt{7}\right)}{18}$$

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

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)^{2}}{18}$$

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

$$\frac{32-10\sqrt{7}}{18}+\frac{25+10\sqrt{7}+\left(\sqrt{7}\right)^{2}}{18}$$

\sqrt{7} kvadrati – 7.

$$\frac{32-10\sqrt{7}}{18}+\frac{25+10\sqrt{7}+7}{18}$$

32 olish uchun 25 va 7'ni qo'shing.

$$\frac{32-10\sqrt{7}}{18}+\frac{32+10\sqrt{7}}{18}$$

\frac{32-10\sqrt{7}}{18} va \frac{32+10\sqrt{7}}{18} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{32-10\sqrt{7}+32+10\sqrt{7}}{18}$$

32-10\sqrt{7}+32+10\sqrt{7} hisob-kitobini qiling.

$$\frac{64}{18}$$

\frac{64}{18} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$\frac{32}{9}$$

Omil

$$\frac{2 ^ {5}}{3 ^ {2}} = 3\frac{5}{9} = 3,5555555555555554$$