(1/sqrt{5-sqrt{2}}+1/sqrt{5+sqrt{4}})^2 eching

$$( \frac { 1 } { \sqrt { 5 } - \sqrt { 2 } } + \frac { 1 } { \sqrt { 5 } + \sqrt { 4 } } ) ^ { 2 }$$

Baholash

$$\frac{8\sqrt{10}}{9}-\frac{4\sqrt{2}}{3}-\frac{16\sqrt{5}}{3}+\frac{118}{9}\approx 2,110710624$$

Qisqa yechim qadamlari

\frac{1}{\sqrt{5}-\sqrt{2}} maxrajini \sqrt{5}+\sqrt{2} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$\left(\frac{\sqrt{5}+\sqrt{2}}{\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{2}\right)}+\frac{1}{\sqrt{5}+\sqrt{4}}\right)^{2}$$

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

$$\left(\frac{\sqrt{5}+\sqrt{2}}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{2}\right)^{2}}+\frac{1}{\sqrt{5}+\sqrt{4}}\right)^{2}$$

\sqrt{5} kvadratini chiqarish. \sqrt{2} kvadratini chiqarish.

$$\left(\frac{\sqrt{5}+\sqrt{2}}{5-2}+\frac{1}{\sqrt{5}+\sqrt{4}}\right)^{2}$$

3 olish uchun 5 dan 2 ni ayirish.

$$\left(\frac{\sqrt{5}+\sqrt{2}}{3}+\frac{1}{\sqrt{5}+\sqrt{4}}\right)^{2}$$

4 ning kvadrat ildizini hisoblab, 2 natijaga ega bo‘ling.

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

\frac{1}{\sqrt{5}+2} maxrajini \sqrt{5}-2 orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

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

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

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

\sqrt{5} kvadratini chiqarish. 2 kvadratini chiqarish.

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

1 olish uchun 5 dan 4 ni ayirish.

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

Har qanday son birga bo‘linganda, natija o‘zi chiqadi.

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

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. \sqrt{5}-2 ni \frac{3}{3} marotabaga ko'paytirish.

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

\frac{\sqrt{5}+\sqrt{2}}{3} va \frac{3\left(\sqrt{5}-2\right)}{3} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

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

\sqrt{5}+\sqrt{2}+3\left(\sqrt{5}-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\left(\frac{\sqrt{5}+\sqrt{2}+3\sqrt{5}-6}{3}\right)^{2}$$

\sqrt{5}+\sqrt{2}+3\sqrt{5}-6 hisob-kitobini qiling.

$$\left(\frac{4\sqrt{5}+\sqrt{2}-6}{3}\right)^{2}$$

\frac{4\sqrt{5}+\sqrt{2}-6}{3}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{\left(4\sqrt{5}+\sqrt{2}-6\right)^{2}}{3^{2}}$$

4\sqrt{5}+\sqrt{2}-6 kvadratini chiqarish.

$$\frac{8\sqrt{2}\sqrt{5}+16\left(\sqrt{5}\right)^{2}+\left(\sqrt{2}\right)^{2}-48\sqrt{5}-12\sqrt{2}+36}{3^{2}}$$

\sqrt{2} va \sqrt{5} ni koʻpaytirish uchun kvadrat ildiz ichidagi sonlarni koʻpaytiring.

$$\frac{8\sqrt{10}+16\left(\sqrt{5}\right)^{2}+\left(\sqrt{2}\right)^{2}-48\sqrt{5}-12\sqrt{2}+36}{3^{2}}$$

\sqrt{5} kvadrati – 5.

$$\frac{8\sqrt{10}+16\times 5+\left(\sqrt{2}\right)^{2}-48\sqrt{5}-12\sqrt{2}+36}{3^{2}}$$

80 hosil qilish uchun 16 va 5 ni ko'paytirish.

$$\frac{8\sqrt{10}+80+\left(\sqrt{2}\right)^{2}-48\sqrt{5}-12\sqrt{2}+36}{3^{2}}$$

\sqrt{2} kvadrati – 2.

$$\frac{8\sqrt{10}+80+2-48\sqrt{5}-12\sqrt{2}+36}{3^{2}}$$

82 olish uchun 80 va 2'ni qo'shing.

$$\frac{8\sqrt{10}+82-48\sqrt{5}-12\sqrt{2}+36}{3^{2}}$$

118 olish uchun 82 va 36'ni qo'shing.

$$\frac{8\sqrt{10}+118-48\sqrt{5}-12\sqrt{2}}{3^{2}}$$

2 daraja ko‘rsatkichini 3 ga hisoblang va 9 ni qiymatni oling.

$$\frac{8\sqrt{10}+118-48\sqrt{5}-12\sqrt{2}}{9}$$

Kengaytirish

$$\frac{8 \sqrt{10}}{9} - \frac{4 \sqrt{2}}{3} - \frac{16 \sqrt{5}}{3} + \frac{118}{9} = 2,110710624$$

Qisqa yechim qadamlari

\frac{1}{\sqrt{5}-\sqrt{2}} maxrajini \sqrt{5}+\sqrt{2} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$\left(\frac{\sqrt{5}+\sqrt{2}}{\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{2}\right)}+\frac{1}{\sqrt{5}+\sqrt{4}}\right)^{2}$$

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

$$\left(\frac{\sqrt{5}+\sqrt{2}}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{2}\right)^{2}}+\frac{1}{\sqrt{5}+\sqrt{4}}\right)^{2}$$

\sqrt{5} kvadratini chiqarish. \sqrt{2} kvadratini chiqarish.

$$\left(\frac{\sqrt{5}+\sqrt{2}}{5-2}+\frac{1}{\sqrt{5}+\sqrt{4}}\right)^{2}$$

3 olish uchun 5 dan 2 ni ayirish.

$$\left(\frac{\sqrt{5}+\sqrt{2}}{3}+\frac{1}{\sqrt{5}+\sqrt{4}}\right)^{2}$$

4 ning kvadrat ildizini hisoblab, 2 natijaga ega bo‘ling.

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

\frac{1}{\sqrt{5}+2} maxrajini \sqrt{5}-2 orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

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

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

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

\sqrt{5} kvadratini chiqarish. 2 kvadratini chiqarish.

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

1 olish uchun 5 dan 4 ni ayirish.

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

Har qanday son birga bo‘linganda, natija o‘zi chiqadi.

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

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. \sqrt{5}-2 ni \frac{3}{3} marotabaga ko'paytirish.

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

\frac{\sqrt{5}+\sqrt{2}}{3} va \frac{3\left(\sqrt{5}-2\right)}{3} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

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

\sqrt{5}+\sqrt{2}+3\left(\sqrt{5}-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\left(\frac{\sqrt{5}+\sqrt{2}+3\sqrt{5}-6}{3}\right)^{2}$$

\sqrt{5}+\sqrt{2}+3\sqrt{5}-6 hisob-kitobini qiling.

$$\left(\frac{4\sqrt{5}+\sqrt{2}-6}{3}\right)^{2}$$

\frac{4\sqrt{5}+\sqrt{2}-6}{3}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{\left(4\sqrt{5}+\sqrt{2}-6\right)^{2}}{3^{2}}$$

4\sqrt{5}+\sqrt{2}-6 kvadratini chiqarish.

$$\frac{8\sqrt{2}\sqrt{5}+16\left(\sqrt{5}\right)^{2}+\left(\sqrt{2}\right)^{2}-48\sqrt{5}-12\sqrt{2}+36}{3^{2}}$$

\sqrt{2} va \sqrt{5} ni koʻpaytirish uchun kvadrat ildiz ichidagi sonlarni koʻpaytiring.

$$\frac{8\sqrt{10}+16\left(\sqrt{5}\right)^{2}+\left(\sqrt{2}\right)^{2}-48\sqrt{5}-12\sqrt{2}+36}{3^{2}}$$

\sqrt{5} kvadrati – 5.

$$\frac{8\sqrt{10}+16\times 5+\left(\sqrt{2}\right)^{2}-48\sqrt{5}-12\sqrt{2}+36}{3^{2}}$$

80 hosil qilish uchun 16 va 5 ni ko'paytirish.

$$\frac{8\sqrt{10}+80+\left(\sqrt{2}\right)^{2}-48\sqrt{5}-12\sqrt{2}+36}{3^{2}}$$

\sqrt{2} kvadrati – 2.

$$\frac{8\sqrt{10}+80+2-48\sqrt{5}-12\sqrt{2}+36}{3^{2}}$$

82 olish uchun 80 va 2'ni qo'shing.

$$\frac{8\sqrt{10}+82-48\sqrt{5}-12\sqrt{2}+36}{3^{2}}$$

118 olish uchun 82 va 36'ni qo'shing.

$$\frac{8\sqrt{10}+118-48\sqrt{5}-12\sqrt{2}}{3^{2}}$$

2 daraja ko‘rsatkichini 3 ga hisoblang va 9 ni qiymatni oling.

$$\frac{8\sqrt{10}+118-48\sqrt{5}-12\sqrt{2}}{9}$$