Question

Zapisz liczbę w postaci potęgi o podstawie 6.
a) [tex]\sqrt[6]{6\sqrt{6} }[/tex]
b) [tex]\sqrt[3]{6\sqrt{36\sqrt{6} } }[/tex]
c) 6 × [tex]\sqrt{6\sqrt[3]{36\sqrt[3]{6} } }[/tex]

Answer 1

[tex]\sqrt[6]{6\sqrt6}=\sqrt[6]{6*6^{\frac{1}{2}}}=\sqrt[6]{6^{1\frac{1}{2}}}=(6^{1\frac{1}{2}})^{\frac{1}{6}}=6^{\frac{3}{2}*\frac{1}{6}}=6^{\frac{1}{2}*\frac{1}{2}}=6^{\frac{1}{4}}[/tex]

[tex]\sqrt[3]{6\sqrt{36\sqrt6}}=\sqrt[3]{6\sqrt{6^2*6^{\frac{1}{2}}}}=\sqrt[3]{6\sqrt{6^{2\frac{1}{2}}}}=\sqrt[3]{6*(6^{2\frac{1}{2}})^{\frac{1}{2}}}=\sqrt[3]{6*6^{\frac{5}{2}*\frac{1}{2}}}=\\=\sqrt[3]{6*6^{\frac{5}{4}}}=\sqrt[3]{6*6^{1\frac{1}{4}}}=\sqrt[3]{6^{2\frac{1}{4}}}=(6^{2\frac{1}{4}})^{\frac{1}{3}}=6^{\frac{9}{4}*\frac{1}{3}}=6^{\frac{3}{4}*\frac{1}{1}}=6^{\frac{3}{4}[/tex]

[tex]6*\sqrt{6\sqrt[3]{36\sqrt[3]6}}=6*\sqrt{6\sqrt[3]{6^2*6^{\frac{1}{3}}}}=6*\sqrt{6\sqrt[3]{6^{2\frac{1}{3}}}}=6*\sqrt{6*(6^{2\frac{1}{3}})^{\frac{1}{3}}}}=\\=6*\sqrt{6*6^{\frac{7}{3}*\frac{1}{3}}}=6*\sqrt{6*6^{\frac{7}{9}}}=6*\sqrt{6^{1\frac{7}{9}}}=6*(6^{1\frac{7}{9}})^{\frac{1}{2}}=6*6^{\frac{16}{9}*\frac{1}{2}}=6*6^{\frac{8}{9}*\frac{1}{1}}=\\=6*6^{\frac{8}{9}}=6^{1\frac{8}{9}}[/tex]