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

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

Nilaikan

$$\frac{25\times 4096^{\frac{23}{24}}\sqrt[20]{9765625}}{1024}\approx 158.113883008$$

Langkah Penyelesaian

Kira 20 dikuasakan 30 dan dapatkan 1073741824000000000000000000000000000000.

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

Kira 3 dikuasakan 20 dan dapatkan 3486784401.

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

Darabkan 1073741824000000000000000000000000000000 dan 3486784401 untuk mendapatkan 3743906242624487424000000000000000000000000000000.

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

Kira 5 dikuasakan 60 dan dapatkan 867361737988403547205962240695953369140625.

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

Darabkan 3743906242624487424000000000000000000000000000000 dan 867361737988403547205962240695953369140625 untuk mendapatkan 3247321025468409061431884765625000000000000000000000000000000000000000000000000000000000000.

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

Kira punca kuasa dua 3247321025468409061431884765625000000000000000000000000000000000000000000000000000000000000 dan dapatkan 1802032470703125000000000000000000000000000000.

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

Kira \sqrt[5]{1802032470703125000000000000000000000000000000} dan dapatkan 1125000000.

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

Kira 2 dikuasakan 36 dan dapatkan 68719476736.

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

Kira 3 dikuasakan 24 dan dapatkan 282429536481.

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

Darabkan 68719476736 dan 282429536481 untuk mendapatkan 19408409961765342806016.

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

Kira 5 dikuasakan 48 dan dapatkan 3552713678800500929355621337890625.

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

Darabkan 19408409961765342806016 dan 3552713678800500929355621337890625 untuk mendapatkan 68952523554931640625000000000000000000000000000000000000.

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

Kira punca kuasa dua 68952523554931640625000000000000000000000000000000000000 dan dapatkan 8303765625000000000000000000.

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

Kira punca kuasa dua 8303765625000000000000000000 dan dapatkan 91125000000000.

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

Kira \sqrt[3]{91125000000000} dan dapatkan 45000.

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

Bahagikan 1125000000 dengan 45000 untuk mendapatkan 25000.

$$\sqrt{25000}$$

Faktor 25000=50^{2}\times 10. Tulis semula punca kuasa dua untuk hasil \sqrt{50^{2}\times 10} sebagai hasil punca kuasa \sqrt{50^{2}}\sqrt{10}. Ambil punca kuasa dua 50^{2}.

$$50\sqrt{10}$$