sqrt{6}*(5)(2sqrt{5}+5sqrt{2})^2-(2sqrt{5}-5sqrt{2})^2 eching

$$\sqrt { 6 } \cdot ( 5 ) ( 2 \sqrt { 5 } + 5 \sqrt { 2 } ) ^ { 2 } - ( 2 \sqrt { 5 } - 5 \sqrt { 2 } ) ^ { 2 }$$

Baholash

$$20\sqrt{10}+200\sqrt{15}+350\sqrt{6}-70\approx 1625,163632419$$

Yechim qadamlari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2\sqrt{5}+5\sqrt{2}\right)^{2} kengaytirilishi uchun ishlating.

$$\sqrt{6}\times 5\left(4\left(\sqrt{5}\right)^{2}+20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

\sqrt{5} kvadrati – 5.

$$\sqrt{6}\times 5\left(4\times 5+20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

20 hosil qilish uchun 4 va 5 ni ko'paytirish.

$$\sqrt{6}\times 5\left(20+20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

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

$$\sqrt{6}\times 5\left(20+20\sqrt{10}+25\left(\sqrt{2}\right)^{2}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

\sqrt{2} kvadrati – 2.

$$\sqrt{6}\times 5\left(20+20\sqrt{10}+25\times 2\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

50 hosil qilish uchun 25 va 2 ni ko'paytirish.

$$\sqrt{6}\times 5\left(20+20\sqrt{10}+50\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

70 olish uchun 20 va 50'ni qo'shing.

$$\sqrt{6}\times 5\left(70+20\sqrt{10}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

\sqrt{6}\times 5 ga 70+20\sqrt{10} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$70\sqrt{6}\times 5+100\sqrt{6}\sqrt{10}-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

350 hosil qilish uchun 70 va 5 ni ko'paytirish.

$$350\sqrt{6}+100\sqrt{6}\sqrt{10}-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

\sqrt{6} va \sqrt{10} ni koʻpaytirish uchun kvadrat ildiz ichidagi sonlarni koʻpaytiring.

$$350\sqrt{6}+100\sqrt{60}-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(2\sqrt{5}-5\sqrt{2}\right)^{2} kengaytirilishi uchun ishlating.

$$350\sqrt{6}+100\sqrt{60}-\left(4\left(\sqrt{5}\right)^{2}-20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)$$

\sqrt{5} kvadrati – 5.

$$350\sqrt{6}+100\sqrt{60}-\left(4\times 5-20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)$$

20 hosil qilish uchun 4 va 5 ni ko'paytirish.

$$350\sqrt{6}+100\sqrt{60}-\left(20-20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)$$

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

$$350\sqrt{6}+100\sqrt{60}-\left(20-20\sqrt{10}+25\left(\sqrt{2}\right)^{2}\right)$$

\sqrt{2} kvadrati – 2.

$$350\sqrt{6}+100\sqrt{60}-\left(20-20\sqrt{10}+25\times 2\right)$$

50 hosil qilish uchun 25 va 2 ni ko'paytirish.

$$350\sqrt{6}+100\sqrt{60}-\left(20-20\sqrt{10}+50\right)$$

70 olish uchun 20 va 50'ni qo'shing.

$$350\sqrt{6}+100\sqrt{60}-\left(70-20\sqrt{10}\right)$$

70-20\sqrt{10} teskarisini topish uchun har birining teskarisini toping.

$$350\sqrt{6}+100\sqrt{60}-70+20\sqrt{10}$$

Faktor: 60=2^{2}\times 15. \sqrt{2^{2}\times 15} koʻpaytmasining kvadrat ildizini \sqrt{2^{2}}\sqrt{15} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 2^{2} ning kvadrat ildizini chiqarish.

$$350\sqrt{6}+100\times 2\sqrt{15}-70+20\sqrt{10}$$

200 hosil qilish uchun 100 va 2 ni ko'paytirish.

$$350\sqrt{6}+200\sqrt{15}-70+20\sqrt{10}$$