frac{sqrt{12}+sqrt{6}+sqrt{2}+2}{sqrt{3}+1} eching

$$\frac { \sqrt { 12 } + \sqrt { 6 } + \sqrt { 2 } + 2 } { \sqrt { 3 } + 1 }$$

Baholash

$$\sqrt{2}+2\approx 3,414213562$$

Yechim qadamlari

Faktor: 12=2^{2}\times 3. \sqrt{2^{2}\times 3} koʻpaytmasining kvadrat ildizini \sqrt{2^{2}}\sqrt{3} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 2^{2} ning kvadrat ildizini chiqarish.

$$\frac{2\sqrt{3}+\sqrt{6}+\sqrt{2}+2}{\sqrt{3}+1}$$

\frac{2\sqrt{3}+\sqrt{6}+\sqrt{2}+2}{\sqrt{3}+1} maxrajini \sqrt{3}-1 orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

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

Hisoblang: \left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

\sqrt{3} kvadratini chiqarish. 1 kvadratini chiqarish.

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

2 olish uchun 3 dan 1 ni ayirish.

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

2\sqrt{3}+\sqrt{6}+\sqrt{2}+2 ifodaning har bir elementini \sqrt{3}-1 ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

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

\sqrt{3} kvadrati – 3.

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

6 hosil qilish uchun 2 va 3 ni ko'paytirish.

$$\frac{6-2\sqrt{3}+\sqrt{6}\sqrt{3}-\sqrt{6}+\sqrt{2}\sqrt{3}-\sqrt{2}+2\sqrt{3}-2}{2}$$

Faktor: 6=3\times 2. \sqrt{3\times 2} koʻpaytmasining kvadrat ildizini \sqrt{3}\sqrt{2} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing.

$$\frac{6-2\sqrt{3}+\sqrt{3}\sqrt{2}\sqrt{3}-\sqrt{6}+\sqrt{2}\sqrt{3}-\sqrt{2}+2\sqrt{3}-2}{2}$$

3 hosil qilish uchun \sqrt{3} va \sqrt{3} ni ko'paytirish.

$$\frac{6-2\sqrt{3}+3\sqrt{2}-\sqrt{6}+\sqrt{2}\sqrt{3}-\sqrt{2}+2\sqrt{3}-2}{2}$$

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

$$\frac{6-2\sqrt{3}+3\sqrt{2}-\sqrt{6}+\sqrt{6}-\sqrt{2}+2\sqrt{3}-2}{2}$$

0 ni olish uchun -\sqrt{6} va \sqrt{6} ni birlashtirish.

$$\frac{6-2\sqrt{3}+3\sqrt{2}-\sqrt{2}+2\sqrt{3}-2}{2}$$

2\sqrt{2} ni olish uchun 3\sqrt{2} va -\sqrt{2} ni birlashtirish.

$$\frac{6-2\sqrt{3}+2\sqrt{2}+2\sqrt{3}-2}{2}$$

0 ni olish uchun -2\sqrt{3} va 2\sqrt{3} ni birlashtirish.

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

4 olish uchun 6 dan 2 ni ayirish.

$$\frac{4+2\sqrt{2}}{2}$$

2+\sqrt{2} natijani olish uchun 4+2\sqrt{2} ning har bir ifodasini 2 ga bo‘ling.

$$2+\sqrt{2}$$