# Evaluate

Question:

Evaluate

(i) $\frac{4}{(216)^{-\frac{2}{3}}}+\frac{1}{(256)^{-\frac{3}{4}}}+\frac{2}{(243)^{-\frac{1}{5}}}$

(ii) $\left(\frac{64}{125}\right)^{-\frac{2}{3}}+\left(\frac{256}{625}\right)^{-\frac{1}{4}}+\left(\frac{3}{7}\right)^{0}$

(iii) $\left(\frac{81}{16}\right)^{-\frac{3}{4}}\left[\left(\frac{25}{9}\right)^{-\frac{3}{2}} \div\left(\frac{5}{2}\right)^{-3}\right]$

(iv) $\frac{(25)^{\frac{5}{2}} \times(729)^{\frac{1}{3}}}{(125)^{\frac{2}{3}} \times(27)^{\frac{2}{3}} \times 8^{\frac{4}{3}}}$

Solution:

(i) $\frac{4}{(216)^{-\frac{2}{3}}}+\frac{1}{(256)^{-\frac{3}{4}}}+\frac{2}{(243)^{-\frac{1}{5}}}$

$\frac{4}{(216)^{-\frac{2}{3}}}+\frac{1}{(256)^{-\frac{3}{4}}}+\frac{2}{(243)^{-\frac{1}{5}}}$

$=\frac{4}{\left[(6)^{3}\right]^{-\frac{2}{3}}}+\frac{1}{\left[(4)^{4}\right]^{-\frac{3}{4}}}+\frac{2}{\left[(3)^{5}\right]^{-\frac{1}{5}}}$

$=\frac{4}{(6)^{-2}}+\frac{1}{(4)^{-3}}+\frac{2}{(3)^{-1}}$

$=4(6)^{2}+(4)^{3}+2(3)$

$=144+64+6$

$=214$

(ii) $\left(\frac{64}{125}\right)^{-\frac{2}{3}}+\left(\frac{256}{625}\right)^{-\frac{1}{4}}+\left(\frac{3}{7}\right)^{0}$

$\left(\frac{64}{125}\right)^{-\frac{2}{3}}+\left(\frac{256}{625}\right)^{-\frac{1}{4}}+\left(\frac{3}{7}\right)^{0}$

$=\left[\left(\frac{4}{5}\right)^{3}\right]^{-\frac{2}{3}}+\left[\left(\frac{4}{5}\right)^{4}\right]^{-\frac{1}{4}}+1$

$=\left(\frac{4}{5}\right)^{-2}+\left(\frac{4}{5}\right)^{-1}+1$

$=\left(\frac{5}{4}\right)^{2}+\left(\frac{5}{4}\right)+1$

$=\frac{25}{16}+\frac{5}{4}+1$

$=\frac{25+20+16}{16}$

$=\frac{61}{16}$

(iii) $\left(\frac{81}{16}\right)^{-\frac{3}{4}}\left[\left(\frac{25}{9}\right)^{-\frac{3}{2}} \div\left(\frac{5}{2}\right)^{-3}\right]$

$\left(\frac{81}{16}\right)^{-\frac{3}{4}}\left[\left(\frac{25}{9}\right)^{-\frac{3}{2}} \div\left(\frac{5}{2}\right)^{-3}\right]$

$=\left(\frac{16}{81}\right)^{\frac{3}{4}}\left[\left(\frac{9}{25}\right)^{\frac{3}{2}} \div\left(\frac{2}{5}\right)^{3}\right]$

$=\left[\left(\frac{2}{3}\right)^{4}\right]^{\frac{3}{4}}\left\{\left[\left(\frac{3}{5}\right)^{2}\right]^{\frac{3}{2}} \div\left(\frac{8}{125}\right)\right\}$

$=\left(\frac{2}{3}\right)^{3}\left[\left(\frac{3}{5}\right)^{3} \div\left(\frac{8}{125}\right)\right]$

$=\frac{\frac{8}{27} \times \frac{27}{125}}{\frac{8}{125}}$

$=1$

(iv) $\frac{(25)^{\frac{5}{2}} \times(729)^{\frac{1}{3}}}{(125)^{\frac{2}{3}} \times(27)^{\frac{2}{3}} \times 8^{\frac{4}{3}}}$

$\frac{(25)^{\frac{5}{2}} \times(729)^{\frac{1}{3}}}{(125)^{\frac{2}{3}} \times(27)^{\frac{2}{3}} \times 8^{\frac{4}{3}}}$

$=\frac{\left[(5)^{2}\right]^{\frac{5}{2}} \times\left[(9)^{3}\right]^{\frac{1}{3}}}{\left[(5)^{3}\right]^{\frac{2}{3}} \times\left[(3)^{3}\right]^{\frac{2}{3}} \times\left[(2)^{3}\right]^{\frac{4}{3}}}$

$=\frac{(5)^{5} \times(9)^{1}}{(5)^{2} \times(3)^{2} \times(2)^{4}}$

$=\frac{5 \times 5 \times 5 \times 5 \times 5 \times 9}{5 \times 5 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2}$

$=\frac{125}{16}$