Answer:
[tex]\sqrt[20]{6}[/tex]
[tex]\sqrt[60]{6^{137} }[/tex]
Step-by-step explanation:
We have to simplify the followings:
1) [tex]\frac{\sqrt[4]{6} }{\sqrt[5]{6} }[/tex] and
2) [tex]6^{\frac{1}{5}} \times 6^{\frac{9}{12}} \times 6^{\frac{1}{12}} \times 6^{\frac{5}{4}}[/tex]
1) Now, [tex]\frac{\sqrt[4]{6} }{\sqrt[5]{6} }[/tex]
= [tex]\frac{6^{\frac{1}{4}}}{6^{\frac{1}{5}}}[/tex]
= [tex]6^{(\frac{1}{4} - \frac{1}{5})}[/tex]
{Since, [tex]\frac{a^{b}}{a^{c}} = a^{(b - c)}[/tex] }
= [tex]6^{(\frac{5 - 4}{20})}[/tex]
= [tex]6^{\frac{1}{20}}[/tex]
= [tex]\sqrt[20]{6}[/tex] (Answer)
2) [tex]6^{\frac{1}{5}} \times 6^{\frac{9}{12}} \times 6^{\frac{1}{12}} \times 6^{\frac{5}{4}}[/tex]
= [tex]6^{(\frac{1}{5} + \frac{9}{12} + \frac{1}{12} + \frac{5}{4})}[/tex]
= [tex]6^{(\frac{12 + 45 + 5 + 75}{60})}[/tex]
{Since, [tex]a^{b} \times a^{c} = a^{(b + c)}[/tex]}
= [tex]6^{\frac{137}{60}}[/tex]
= [tex]\sqrt[60]{6^{137} }[/tex] (Answer)
