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

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

Evaluer

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

Løsningsvejledning

Beregn 20 til potensen af 30, og få 1073741824000000000000000000000000000000.

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

Beregn 3 til potensen af 20, og få 3486784401.

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

Multiplicer 1073741824000000000000000000000000000000 og 3486784401 for at få 3743906242624487424000000000000000000000000000000.

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

Beregn 5 til potensen af 60, og få 867361737988403547205962240695953369140625.

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

Multiplicer 3743906242624487424000000000000000000000000000000 og 867361737988403547205962240695953369140625 for at få 3247321025468409061431884765625000000000000000000000000000000000000000000000000000000000000.

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

Beregn kvadratroden af 3247321025468409061431884765625000000000000000000000000000000000000000000000000000000000000, og find 1802032470703125000000000000000000000000000000.

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

Beregn \sqrt[5]{1802032470703125000000000000000000000000000000}, og find 1125000000.

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

Beregn 2 til potensen af 36, og få 68719476736.

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

Beregn 3 til potensen af 24, og få 282429536481.

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

Multiplicer 68719476736 og 282429536481 for at få 19408409961765342806016.

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

Beregn 5 til potensen af 48, og få 3552713678800500929355621337890625.

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

Multiplicer 19408409961765342806016 og 3552713678800500929355621337890625 for at få 68952523554931640625000000000000000000000000000000000000.

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

Beregn kvadratroden af 68952523554931640625000000000000000000000000000000000000, og find 8303765625000000000000000000.

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

Beregn kvadratroden af 8303765625000000000000000000, og find 91125000000000.

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

Beregn \sqrt[3]{91125000000000}, og find 45000.

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

Divider 1125000000 med 45000 for at få 25000.

$$\sqrt{25000}$$

Faktoriser 25000=50^{2}\times 10. Omskriv kvadratroden af produktet \sqrt{50^{2}\times 10} som produktet af kvadratrødderne \sqrt{50^{2}}\sqrt{10}. Tag kvadratroden af 50^{2}.

$$50\sqrt{10}$$