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

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

Laske

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

Ratkaisun vaiheet

Laske 20 potenssiin 30, jolloin ratkaisuksi tulee 1073741824000000000000000000000000000000.

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

Laske 3 potenssiin 20, jolloin ratkaisuksi tulee 3486784401.

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

Kerro 1073741824000000000000000000000000000000 ja 3486784401, niin saadaan 3743906242624487424000000000000000000000000000000.

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

Laske 5 potenssiin 60, jolloin ratkaisuksi tulee 867361737988403547205962240695953369140625.

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

Kerro 3743906242624487424000000000000000000000000000000 ja 867361737988403547205962240695953369140625, niin saadaan 3247321025468409061431884765625000000000000000000000000000000000000000000000000000000000000.

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

Laske luvun 3247321025468409061431884765625000000000000000000000000000000000000000000000000000000000000 neliöjuuri, saat vastaukseksi 1802032470703125000000000000000000000000000000.

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

Laske \sqrt[5]{1802032470703125000000000000000000000000000000}, saat tulokseksi 1125000000.

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

Laske 2 potenssiin 36, jolloin ratkaisuksi tulee 68719476736.

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

Laske 3 potenssiin 24, jolloin ratkaisuksi tulee 282429536481.

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

Kerro 68719476736 ja 282429536481, niin saadaan 19408409961765342806016.

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

Laske 5 potenssiin 48, jolloin ratkaisuksi tulee 3552713678800500929355621337890625.

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

Kerro 19408409961765342806016 ja 3552713678800500929355621337890625, niin saadaan 68952523554931640625000000000000000000000000000000000000.

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

Laske luvun 68952523554931640625000000000000000000000000000000000000 neliöjuuri, saat vastaukseksi 8303765625000000000000000000.

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

Laske luvun 8303765625000000000000000000 neliöjuuri, saat vastaukseksi 91125000000000.

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

Laske \sqrt[3]{91125000000000}, saat tulokseksi 45000.

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

Jaa 1125000000 luvulla 45000, jolloin ratkaisuksi tulee 25000.

$$\sqrt{25000}$$

Jaa 25000=50^{2}\times 10 tekijöihin. Kirjoita tuotteen neliö juuri \sqrt{50^{2}\times 10} neliö juuren tulo \sqrt{50^{2}}\sqrt{10}. Ota luvun 50^{2} neliöjuuri.

$$50\sqrt{10}$$