Answer
$\sqrt[6]{243}$
Work Step by Step
RECALL:
(i) $\sqrt[n]{a} = a^{\frac{1}{n}}$
(ii) $\sqrt[n]{a^m} = a^{\frac{m}{n}}$
(iii) $a^{\frac{m}{n}}=\sqrt[n]{a^m}$
(iv) $a^m \cdot a^n = a^{m+n}$
Use rule (i) above to obtain:
$=3^{\frac{1}{2}} \cdot 3^{\frac{1}{3}}$
Use rule (iv) above to obtain:
$=3^{\frac{1}{2}+\frac{1}{3}}
\\=3^{\frac{3}{6} + \frac{2}{6}}
\\=3^{\frac{5}{6}}$
Use rule (iii) above, with m=5 and n=6, to obtain:
$\\=\sqrt[6]{3^5}
\\=\sqrt[6]{3(3)(3)(3)(3)}
\\=\sqrt[6]{243}$
