frac{1/sqrt{5+sqrt{3}}}{1/sqrt{5}-sqrt{3}} eching

$$\frac { \frac { 1 } { \sqrt { 5 } + \sqrt { 3 } } } { \frac { 1 } { \sqrt { 5 } } - \sqrt { 3 } }$$

Baholash

$$-\frac{\sqrt{15}}{7}+\frac{5}{14}\approx -0,196140478$$

Qisqa yechim qadamlari

\frac{1}{\sqrt{5}+\sqrt{3}} maxrajini \sqrt{5}-\sqrt{3} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$\frac{\frac{\sqrt{5}-\sqrt{3}}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}}{\frac{1}{\sqrt{5}}-\sqrt{3}}$$

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}.

$$\frac{\frac{\sqrt{5}-\sqrt{3}}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}}{\frac{1}{\sqrt{5}}-\sqrt{3}}$$

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

$$\frac{\frac{\sqrt{5}-\sqrt{3}}{5-3}}{\frac{1}{\sqrt{5}}-\sqrt{3}}$$

2 olish uchun 5 dan 3 ni ayirish.

$$\frac{\frac{\sqrt{5}-\sqrt{3}}{2}}{\frac{1}{\sqrt{5}}-\sqrt{3}}$$

\frac{1}{\sqrt{5}} maxrajini \sqrt{5} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

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

\sqrt{5} kvadrati – 5.

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

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. \sqrt{3} ni \frac{5}{5} marotabaga ko'paytirish.

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

\frac{\sqrt{5}}{5} va \frac{5\sqrt{3}}{5} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

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

\frac{\sqrt{5}-\sqrt{3}}{2} ni \frac{\sqrt{5}-5\sqrt{3}}{5} ga bo'lish \frac{\sqrt{5}-\sqrt{3}}{2} ga k'paytirish \frac{\sqrt{5}-5\sqrt{3}}{5} ga qaytarish.

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

\sqrt{5}-\sqrt{3} ga 5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

2 ga \sqrt{5}-5\sqrt{3} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{5\sqrt{5}-5\sqrt{3}}{2\sqrt{5}-10\sqrt{3}}$$

\frac{5\sqrt{5}-5\sqrt{3}}{2\sqrt{5}-10\sqrt{3}} maxrajini 2\sqrt{5}+10\sqrt{3} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{\left(2\sqrt{5}-10\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}$$

Hisoblang: \left(2\sqrt{5}-10\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{\left(2\sqrt{5}\right)^{2}-\left(-10\sqrt{3}\right)^{2}}$$

\left(2\sqrt{5}\right)^{2} ni kengaytirish.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{2^{2}\left(\sqrt{5}\right)^{2}-\left(-10\sqrt{3}\right)^{2}}$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{4\left(\sqrt{5}\right)^{2}-\left(-10\sqrt{3}\right)^{2}}$$

\sqrt{5} kvadrati – 5.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{4\times 5-\left(-10\sqrt{3}\right)^{2}}$$

20 hosil qilish uchun 4 va 5 ni ko'paytirish.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{20-\left(-10\sqrt{3}\right)^{2}}$$

\left(-10\sqrt{3}\right)^{2} ni kengaytirish.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{20-\left(-10\right)^{2}\left(\sqrt{3}\right)^{2}}$$

2 daraja ko‘rsatkichini -10 ga hisoblang va 100 ni qiymatni oling.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{20-100\left(\sqrt{3}\right)^{2}}$$

\sqrt{3} kvadrati – 3.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{20-100\times 3}$$

300 hosil qilish uchun 100 va 3 ni ko'paytirish.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{20-300}$$

-280 olish uchun 20 dan 300 ni ayirish.

$$\frac{\left(5\sqrt{5}-5\sqrt{3}\right)\left(2\sqrt{5}+10\sqrt{3}\right)}{-280}$$

5\sqrt{5}-5\sqrt{3} ifodaning har bir elementini 2\sqrt{5}+10\sqrt{3} ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$\frac{10\left(\sqrt{5}\right)^{2}+50\sqrt{3}\sqrt{5}-10\sqrt{3}\sqrt{5}-50\left(\sqrt{3}\right)^{2}}{-280}$$

\sqrt{5} kvadrati – 5.

$$\frac{10\times 5+50\sqrt{3}\sqrt{5}-10\sqrt{3}\sqrt{5}-50\left(\sqrt{3}\right)^{2}}{-280}$$

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

$$\frac{50+50\sqrt{3}\sqrt{5}-10\sqrt{3}\sqrt{5}-50\left(\sqrt{3}\right)^{2}}{-280}$$

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

$$\frac{50+50\sqrt{15}-10\sqrt{3}\sqrt{5}-50\left(\sqrt{3}\right)^{2}}{-280}$$

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

$$\frac{50+50\sqrt{15}-10\sqrt{15}-50\left(\sqrt{3}\right)^{2}}{-280}$$

40\sqrt{15} ni olish uchun 50\sqrt{15} va -10\sqrt{15} ni birlashtirish.

$$\frac{50+40\sqrt{15}-50\left(\sqrt{3}\right)^{2}}{-280}$$

\sqrt{3} kvadrati – 3.

$$\frac{50+40\sqrt{15}-50\times 3}{-280}$$

-150 hosil qilish uchun -50 va 3 ni ko'paytirish.

$$\frac{50+40\sqrt{15}-150}{-280}$$

-100 olish uchun 50 dan 150 ni ayirish.

$$\frac{-100+40\sqrt{15}}{-280}$$