frac{2+sqrt{3}}{2-sqrt{3}}-frac{2-sqrt{3}}{2+sqrt{3}} eching

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

Baholash

$$8\sqrt{3}\approx 13,856406461$$

Yechim qadamlari

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

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

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

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

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

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

1 olish uchun 4 dan 3 ni ayirish.

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

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

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

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

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

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

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

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

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

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

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

1 olish uchun 4 dan 3 ni ayirish.

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

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

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

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

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

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

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

\sqrt{3} kvadrati – 3.

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

7 olish uchun 4 va 3'ni qo'shing.

$$7+4\sqrt{3}-\left(2-\sqrt{3}\right)^{2}$$

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

$$7+4\sqrt{3}-\left(4-4\sqrt{3}+\left(\sqrt{3}\right)^{2}\right)$$

\sqrt{3} kvadrati – 3.

$$7+4\sqrt{3}-\left(4-4\sqrt{3}+3\right)$$

7 olish uchun 4 va 3'ni qo'shing.

$$7+4\sqrt{3}-\left(7-4\sqrt{3}\right)$$

7-4\sqrt{3} teskarisini topish uchun har birining teskarisini toping.

$$7+4\sqrt{3}-7-\left(-4\sqrt{3}\right)$$

-4\sqrt{3} ning teskarisi 4\sqrt{3} ga teng.

$$7+4\sqrt{3}-7+4\sqrt{3}$$

0 olish uchun 7 dan 7 ni ayirish.

$$4\sqrt{3}+4\sqrt{3}$$

8\sqrt{3} ni olish uchun 4\sqrt{3} va 4\sqrt{3} ni birlashtirish.

$$8\sqrt{3}$$