# Simplify

Question:

Simplify

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

(ii) $(14641)^{0.25}$

(iii) $\left(\frac{32}{243}\right)^{-\frac{4}{5}}$

(iv) $\left(\frac{7776}{243}\right)^{-\frac{3}{5}}$

Solution:

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

$\left(\frac{81}{49}\right)^{-\frac{3}{2}}=\left[\left(\frac{9}{7}\right)^{2}\right]^{-\frac{3}{2}}$

$=\left(\frac{9}{7}\right)^{-3}$

$=\left(\frac{7}{9}\right)^{3}$

$=\frac{343}{729}$

(ii) $(14641)^{0.25}$

$(14641)^{0.25}=(14641)^{\frac{25}{100}}$

$=(14641)^{\frac{1}{4}}$

$=\left[(11)^{4}\right]^{\frac{1}{4}}$

$=11$

(iii) $\left(\frac{32}{243}\right)^{-\frac{4}{5}}$

$\left(\frac{32}{243}\right)^{-\frac{4}{5}}=\left(\frac{243}{32}\right)^{\frac{4}{5}}$

$=\left[\left(\frac{3}{2}\right)^{5}\right]^{\frac{4}{5}}$

$=\left(\frac{3}{2}\right)^{4}$

$=\frac{81}{16}$

(iv) $\left(\frac{7776}{243}\right)^{-\frac{3}{5}}$

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

$=\left[\left(\frac{3}{6}\right)^{5}\right]^{\frac{3}{5}}$

$=\left(\frac{1}{2}\right)^{3}$

$=\frac{1}{8}$