sqrt{frac{sqrt[5]{sqrt{20^{30}*3^{20}*5^60}}}{sqrt[3]{sqrt{sqrt{2^36*3^24*5^48}}}}} eching

$$\sqrt { \frac { \sqrt[ 5 ] { \sqrt { 20 ^ { 30 } \cdot 3 ^ { 20 } \cdot 5 ^ { 60 } } } } { \sqrt[ 3 ] { \sqrt { \sqrt { 2 ^ { 36 } \cdot 3 ^ { 24 } \cdot 5 ^ { 48 } } } } } }$$

Baholash

$$\frac{25\times 4096^{\frac{23}{24}}\sqrt[20]{9765625}}{1024}\approx 158,113883008$$

Yechim qadamlari

30 daraja ko‘rsatkichini 20 ga hisoblang va 1073741824000000000000000000000000000000 ni qiymatni oling.

$$\sqrt{\frac{\sqrt[5]{\sqrt{1073741824000000000000000000000000000000\times 3^{20}\times 5^{60}}}}{\sqrt[3]{\sqrt{\sqrt{2^{36}\times 3^{24}\times 5^{48}}}}}}$$

20 daraja ko‘rsatkichini 3 ga hisoblang va 3486784401 ni qiymatni oling.

$$\sqrt{\frac{\sqrt[5]{\sqrt{1073741824000000000000000000000000000000\times 3486784401\times 5^{60}}}}{\sqrt[3]{\sqrt{\sqrt{2^{36}\times 3^{24}\times 5^{48}}}}}}$$

3743906242624487424000000000000000000000000000000 hosil qilish uchun 1073741824000000000000000000000000000000 va 3486784401 ni ko'paytirish.

$$\sqrt{\frac{\sqrt[5]{\sqrt{3743906242624487424000000000000000000000000000000\times 5^{60}}}}{\sqrt[3]{\sqrt{\sqrt{2^{36}\times 3^{24}\times 5^{48}}}}}}$$

60 daraja ko‘rsatkichini 5 ga hisoblang va 867361737988403547205962240695953369140625 ni qiymatni oling.

$$\sqrt{\frac{\sqrt[5]{\sqrt{3743906242624487424000000000000000000000000000000\times 867361737988403547205962240695953369140625}}}{\sqrt[3]{\sqrt{\sqrt{2^{36}\times 3^{24}\times 5^{48}}}}}}$$

3247321025468409061431884765625000000000000000000000000000000000000000000000000000000000000 hosil qilish uchun 3743906242624487424000000000000000000000000000000 va 867361737988403547205962240695953369140625 ni ko'paytirish.

$$\sqrt{\frac{\sqrt[5]{\sqrt{3247321025468409061431884765625000000000000000000000000000000000000000000000000000000000000}}}{\sqrt[3]{\sqrt{\sqrt{2^{36}\times 3^{24}\times 5^{48}}}}}}$$

3247321025468409061431884765625000000000000000000000000000000000000000000000000000000000000 ning kvadrat ildizini hisoblab, 1802032470703125000000000000000000000000000000 natijaga ega bo‘ling.

$$\sqrt{\frac{\sqrt[5]{1802032470703125000000000000000000000000000000}}{\sqrt[3]{\sqrt{\sqrt{2^{36}\times 3^{24}\times 5^{48}}}}}}$$

\sqrt[5]{1802032470703125000000000000000000000000000000} ni hisoblab, 1125000000 natijasiga ega bo‘ling.

$$\sqrt{\frac{1125000000}{\sqrt[3]{\sqrt{\sqrt{2^{36}\times 3^{24}\times 5^{48}}}}}}$$

36 daraja ko‘rsatkichini 2 ga hisoblang va 68719476736 ni qiymatni oling.

$$\sqrt{\frac{1125000000}{\sqrt[3]{\sqrt{\sqrt{68719476736\times 3^{24}\times 5^{48}}}}}}$$

24 daraja ko‘rsatkichini 3 ga hisoblang va 282429536481 ni qiymatni oling.

$$\sqrt{\frac{1125000000}{\sqrt[3]{\sqrt{\sqrt{68719476736\times 282429536481\times 5^{48}}}}}}$$

19408409961765342806016 hosil qilish uchun 68719476736 va 282429536481 ni ko'paytirish.

$$\sqrt{\frac{1125000000}{\sqrt[3]{\sqrt{\sqrt{19408409961765342806016\times 5^{48}}}}}}$$

48 daraja ko‘rsatkichini 5 ga hisoblang va 3552713678800500929355621337890625 ni qiymatni oling.

$$\sqrt{\frac{1125000000}{\sqrt[3]{\sqrt{\sqrt{19408409961765342806016\times 3552713678800500929355621337890625}}}}}$$

68952523554931640625000000000000000000000000000000000000 hosil qilish uchun 19408409961765342806016 va 3552713678800500929355621337890625 ni ko'paytirish.

$$\sqrt{\frac{1125000000}{\sqrt[3]{\sqrt{\sqrt{68952523554931640625000000000000000000000000000000000000}}}}}$$

68952523554931640625000000000000000000000000000000000000 ning kvadrat ildizini hisoblab, 8303765625000000000000000000 natijaga ega bo‘ling.

$$\sqrt{\frac{1125000000}{\sqrt[3]{\sqrt{8303765625000000000000000000}}}}$$

8303765625000000000000000000 ning kvadrat ildizini hisoblab, 91125000000000 natijaga ega bo‘ling.

$$\sqrt{\frac{1125000000}{\sqrt[3]{91125000000000}}}$$

\sqrt[3]{91125000000000} ni hisoblab, 45000 natijasiga ega bo‘ling.

$$\sqrt{\frac{1125000000}{45000}}$$

25000 ni olish uchun 1125000000 ni 45000 ga bo‘ling.

$$\sqrt{25000}$$

Faktor: 25000=50^{2}\times 10. \sqrt{50^{2}\times 10} koʻpaytmasining kvadrat ildizini \sqrt{50^{2}}\sqrt{10} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 50^{2} ning kvadrat ildizini chiqarish.

$$50\sqrt{10}$$