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

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

Izračunaj

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

Koraci rješenja

Izračunajte koliko je 30 na 20 da biste dobili 1073741824000000000000000000000000000000.

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

Izračunajte koliko je 20 na 3 da biste dobili 3486784401.

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

Pomnožite 1073741824000000000000000000000000000000 i 3486784401 da biste dobili 3743906242624487424000000000000000000000000000000.

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

Izračunajte koliko je 60 na 5 da biste dobili 867361737988403547205962240695953369140625.

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

Pomnožite 3743906242624487424000000000000000000000000000000 i 867361737988403547205962240695953369140625 da biste dobili 3247321025468409061431884765625000000000000000000000000000000000000000000000000000000000000.

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

Izračunajte 2. korijen od 3247321025468409061431884765625000000000000000000000000000000000000000000000000000000000000 da biste dobili 1802032470703125000000000000000000000000000000.

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

Izračunajte \sqrt[5]{1802032470703125000000000000000000000000000000} da biste dobili 1125000000.

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

Izračunajte koliko je 36 na 2 da biste dobili 68719476736.

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

Izračunajte koliko je 24 na 3 da biste dobili 282429536481.

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

Pomnožite 68719476736 i 282429536481 da biste dobili 19408409961765342806016.

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

Izračunajte koliko je 48 na 5 da biste dobili 3552713678800500929355621337890625.

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

Pomnožite 19408409961765342806016 i 3552713678800500929355621337890625 da biste dobili 68952523554931640625000000000000000000000000000000000000.

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

Izračunajte 2. korijen od 68952523554931640625000000000000000000000000000000000000 da biste dobili 8303765625000000000000000000.

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

Izračunajte 2. korijen od 8303765625000000000000000000 da biste dobili 91125000000000.

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

Izračunajte \sqrt[3]{91125000000000} da biste dobili 45000.

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

Podijelite 1125000000 s 45000 da biste dobili 25000.

$$\sqrt{25000}$$

Rastavite 25000=50^{2}\times 10 na faktore. Ponovno napišite kvadratni korijen proizvoda \sqrt{50^{2}\times 10} kao umnožak kvadrata korijena \sqrt{50^{2}}\sqrt{10}. Izračunajte kvadratni korijen od 50^{2}.

$$50\sqrt{10}$$