2/sqrt{2-2}+frac{sqrt{2}+1}{sqrt{2}-1}-frac{sqrt{32}}{2} eching

$$\frac { 2 } { \sqrt { 2 } - 2 } + \frac { \sqrt { 2 } + 1 } { \sqrt { 2 } - 1 } - \frac { \sqrt { 32 } } { 2 }$$

Baholash

$$1-\sqrt{2}\approx -0,414213562$$

Yechim qadamlari

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

$$\frac{2\left(\sqrt{2}+2\right)}{\left(\sqrt{2}-2\right)\left(\sqrt{2}+2\right)}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

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

$$\frac{2\left(\sqrt{2}+2\right)}{\left(\sqrt{2}\right)^{2}-2^{2}}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

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

$$\frac{2\left(\sqrt{2}+2\right)}{2-4}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

-2 olish uchun 2 dan 4 ni ayirish.

$$\frac{2\left(\sqrt{2}+2\right)}{-2}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

-2 va -2 ni qisqartiring.

$$-\left(\sqrt{2}+2\right)+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

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

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}-\frac{\sqrt{32}}{2}$$

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

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}-\frac{\sqrt{32}}{2}$$

\sqrt{2} kvadratini chiqarish. 1 kvadratini chiqarish.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{2-1}-\frac{\sqrt{32}}{2}$$

1 olish uchun 2 dan 1 ni ayirish.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{1}-\frac{\sqrt{32}}{2}$$

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

$$-\left(\sqrt{2}+2\right)+\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)-\frac{\sqrt{32}}{2}$$

\left(\sqrt{2}+1\right)^{2} hosil qilish uchun \sqrt{2}+1 va \sqrt{2}+1 ni ko'paytirish.

$$-\left(\sqrt{2}+2\right)+\left(\sqrt{2}+1\right)^{2}-\frac{\sqrt{32}}{2}$$

Faktor: 32=4^{2}\times 2. \sqrt{4^{2}\times 2} koʻpaytmasining kvadrat ildizini \sqrt{4^{2}}\sqrt{2} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 4^{2} ning kvadrat ildizini chiqarish.

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

2\sqrt{2} ni olish uchun 4\sqrt{2} ni 2 ga bo‘ling.

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

\sqrt{2}+2 teskarisini topish uchun har birining teskarisini toping.

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(\sqrt{2}+1\right)^{2} kengaytirilishi uchun ishlating.

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

\sqrt{2} kvadrati – 2.

$$-\sqrt{2}-2+2+2\sqrt{2}+1-2\sqrt{2}$$

3 olish uchun 2 va 1'ni qo'shing.

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

1 olish uchun -2 va 3'ni qo'shing.

$$-\sqrt{2}+1+2\sqrt{2}-2\sqrt{2}$$

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

$$\sqrt{2}+1-2\sqrt{2}$$

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

$$-\sqrt{2}+1$$

Omil

$$1-\sqrt{2}$$