Question

Simplify
(i) $(\frac{81}{49}{)}^{\frac{-3}{2}}$
(ii) $(14641{)}^{0.25}$
(iii) $(\frac{32}{243}{)}^{\frac{-4}{5}}$
(iv) $(\frac{7776}{243}{)}^{\frac{-3}{5}}$

Answer 1

(i) $(\frac{81}{49}{)}^{\frac{-3}{2}}$

Using these identities, $(a^m{)}^n=a^{m\times n},a^{-1}=\frac{1}{a},(\frac{a}{b}{)}^m=\frac{a^m}{b^m}$

$\left(\right(\frac{81}{49}{)}^{\frac{-3}{2}}=\left\{\right(\frac{9}{7}{)}^2{\}}^{\frac{-3}{2}}$ $=(\frac{9}{7}{)}^{-3}=(\frac{7}{9}{)}^3=\frac{343}{729}$
(ii) $(14641{)}^{0.25}$

$(14641{)}^{0.25}=(14641{)}^{\frac{25}{100}}=(14641{)}^{\frac{1}{4}}=({11}^4{)}^{\frac{1}{4}}=11$
(iii) $(\frac{32}{243}{)}^{\frac{-4}{5}}$

$(\frac{32}{243}{)}^{\frac{-4}{5}}=(\frac{243}{32}{)}^{\frac{4}{5}}=\left\{\right(\frac{3}{2}{)}^5{\}}^{\frac{4}{5}}=(\frac{3}{2}{)}^4=\frac{81}{16}$
(iv) $(\frac{7776}{243}{)}^{\frac{-3}{5}}$

$(\frac{7776}{243}{)}^{\frac{-3}{5}}=(\frac{243}{7776}{)}^{\frac{3}{5}}=\left\{\right(\frac{3}{6}{)}^5{\}}^{\frac{3}{5}}=(\frac{1}{2}{)}^3=\frac{1}{8}$