frac{{left(sqrt{2}right)}^2+{left(frac{sqrt{6}+sqrt{2}}{2}right)}^2-{left(sqrt{3}right)}^2}{2*sqrt{2}*(frac{sqrt{6}+sqrt{2}}{2})} eching

$$\frac{ { \left( \sqrt{ 2 } \right) }^{ 2 } + { \left( \frac{ \sqrt{ 6 } + \sqrt{ 2 } }{ 2 } \right) }^{ 2 } - { \left( \sqrt{ 3 } \right) }^{ 2 } }{ 2 \times \sqrt{ 2 } \times ( \frac{ \sqrt{ 6 } + \sqrt{ 2 } }{ 2 } ) }$$

Baholash

$$\frac{1}{2}=0,5$$

Qisqa yechim qadamlari

\sqrt{2} kvadrati – 2.

$$\frac{2+\left(\frac{\sqrt{6}+\sqrt{2}}{2}\right)^{2}-\left(\sqrt{3}\right)^{2}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

\frac{\sqrt{6}+\sqrt{2}}{2}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{2+\frac{\left(\sqrt{6}+\sqrt{2}\right)^{2}}{2^{2}}-\left(\sqrt{3}\right)^{2}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 2 ni \frac{2^{2}}{2^{2}} marotabaga ko'paytirish.

$$\frac{\frac{2\times 2^{2}}{2^{2}}+\frac{\left(\sqrt{6}+\sqrt{2}\right)^{2}}{2^{2}}-\left(\sqrt{3}\right)^{2}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

\frac{2\times 2^{2}}{2^{2}} va \frac{\left(\sqrt{6}+\sqrt{2}\right)^{2}}{2^{2}} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{\frac{2\times 2^{2}+\left(\sqrt{6}+\sqrt{2}\right)^{2}}{2^{2}}-\left(\sqrt{3}\right)^{2}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

2\times 2^{2}+\left(\sqrt{6}+\sqrt{2}\right)^{2} ichidagi ko‘paytirishlarni bajaring.

$$\frac{\frac{8+\left(\sqrt{6}\right)^{2}+2\sqrt{6}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-\left(\sqrt{3}\right)^{2}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

8+\left(\sqrt{6}\right)^{2}+2\sqrt{6}\sqrt{2}+\left(\sqrt{2}\right)^{2} hisob-kitobini qiling.

$$\frac{\frac{16+4\sqrt{3}}{2^{2}}-\left(\sqrt{3}\right)^{2}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

\sqrt{3} kvadrati – 3.

$$\frac{\frac{16+4\sqrt{3}}{2^{2}}-3}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 3 ni \frac{2^{2}}{2^{2}} marotabaga ko'paytirish.

$$\frac{\frac{16+4\sqrt{3}}{2^{2}}-\frac{3\times 2^{2}}{2^{2}}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

\frac{16+4\sqrt{3}}{2^{2}} va \frac{3\times 2^{2}}{2^{2}} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{\frac{16+4\sqrt{3}-3\times 2^{2}}{2^{2}}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

16+4\sqrt{3}-3\times 2^{2} ichidagi ko‘paytirishlarni bajaring.

$$\frac{\frac{16+4\sqrt{3}-12}{2^{2}}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

16+4\sqrt{3}-12 hisob-kitobini qiling.

$$\frac{\frac{4+4\sqrt{3}}{2^{2}}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

2 va 2 ni qisqartiring.

$$\frac{\frac{4+4\sqrt{3}}{2^{2}}}{\left(\sqrt{6}+\sqrt{2}\right)\sqrt{2}}$$

\frac{\frac{4+4\sqrt{3}}{2^{2}}}{\left(\sqrt{6}+\sqrt{2}\right)\sqrt{2}} maxrajini \sqrt{2} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$\frac{\frac{4+4\sqrt{3}}{2^{2}}\sqrt{2}}{\left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{2}\right)^{2}}$$

\sqrt{2} kvadrati – 2.

$$\frac{\frac{4+4\sqrt{3}}{2^{2}}\sqrt{2}}{\left(\sqrt{6}+\sqrt{2}\right)\times 2}$$

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

$$\frac{\frac{4+4\sqrt{3}}{4}\sqrt{2}}{\left(\sqrt{6}+\sqrt{2}\right)\times 2}$$

1+\sqrt{3} natijani olish uchun 4+4\sqrt{3} ning har bir ifodasini 4 ga bo‘ling.

$$\frac{\left(1+\sqrt{3}\right)\sqrt{2}}{\left(\sqrt{6}+\sqrt{2}\right)\times 2}$$

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

$$\frac{\sqrt{2}+\sqrt{3}\sqrt{2}}{\left(\sqrt{6}+\sqrt{2}\right)\times 2}$$

\sqrt{3} va \sqrt{2} ni koʻpaytirish uchun kvadrat ildiz ichidagi sonlarni koʻpaytiring.

$$\frac{\sqrt{2}+\sqrt{6}}{\left(\sqrt{6}+\sqrt{2}\right)\times 2}$$

\sqrt{6}+\sqrt{2} ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{\sqrt{2}+\sqrt{6}}{2\sqrt{6}+2\sqrt{2}}$$

\frac{\sqrt{2}+\sqrt{6}}{2\sqrt{6}+2\sqrt{2}} maxrajini 2\sqrt{6}-2\sqrt{2} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

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

Hisoblang: \left(2\sqrt{6}+2\sqrt{2}\right)\left(2\sqrt{6}-2\sqrt{2}\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

\left(2\sqrt{6}\right)^{2} ni kengaytirish.

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

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

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{4\left(\sqrt{6}\right)^{2}-\left(2\sqrt{2}\right)^{2}}$$

\sqrt{6} kvadrati – 6.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{4\times 6-\left(2\sqrt{2}\right)^{2}}$$

24 hosil qilish uchun 4 va 6 ni ko'paytirish.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{24-\left(2\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{24-2^{2}\left(\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{24-4\left(\sqrt{2}\right)^{2}}$$

\sqrt{2} kvadrati – 2.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{24-4\times 2}$$

8 hosil qilish uchun 4 va 2 ni ko'paytirish.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{24-8}$$

16 olish uchun 24 dan 8 ni ayirish.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{16}$$

\sqrt{2}+\sqrt{6} ga 2\sqrt{6}-2\sqrt{2} ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\frac{-2\left(\sqrt{2}\right)^{2}+2\left(\sqrt{6}\right)^{2}}{16}$$

\sqrt{2} kvadrati – 2.

$$\frac{-2\times 2+2\left(\sqrt{6}\right)^{2}}{16}$$

-4 hosil qilish uchun -2 va 2 ni ko'paytirish.

$$\frac{-4+2\left(\sqrt{6}\right)^{2}}{16}$$

\sqrt{6} kvadrati – 6.

$$\frac{-4+2\times 6}{16}$$

12 hosil qilish uchun 2 va 6 ni ko'paytirish.

$$\frac{-4+12}{16}$$

8 olish uchun -4 va 12'ni qo'shing.

$$\frac{8}{16}$$

\frac{8}{16} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$\frac{1}{2}$$

Omil

$$\frac{1}{2} = 0,5$$