# Evaluate:

Question:

Evaluate:

(i) $\left(\frac{5}{9}\right)^{-2} \times\left(\frac{3}{5}\right)^{-3} \times\left(\frac{3}{5}\right)^{0}$

(ii) $\left(\frac{-3}{5}\right)^{-4} \times\left(\frac{-2}{5}\right)^{2}$

(iii) $\left(\frac{-2}{3}\right)^{-3} \times\left(\frac{-2}{3}\right)^{-2}$

Solution:

(i) $\left(\frac{5}{9}\right)^{-2} \times\left(\frac{3}{5}\right)^{-3} \times\left(\frac{3}{5}\right)^{0}=\left(\frac{5}{9}\right)^{-2} \times\left(\frac{3}{5}\right)^{-3+0}$

$=\left(\frac{5}{9}\right)^{-2} \times\left(\frac{3}{5}\right)^{-3}=\left(\frac{9}{5}\right)^{2} \times\left(\frac{5}{3}\right)^{3}$

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

$=\frac{\left(3^{2}\right)^{2}}{5^{2}} \times \frac{5^{3}}{3^{3}}$

$=\frac{3^{4}}{5^{2}} \times \frac{5^{3}}{3^{3}}=\left(3^{(4-3)}\right) \times\left(5^{(3-2)}\right)=3 \times 5=15$

(ii) $\left(\frac{-3}{5}\right)^{-4} \times\left(\frac{-2}{5}\right)^{2}=\left(\frac{5}{-3}\right)^{4} \times\left(\frac{-2}{5}\right)^{2}$

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

$=25 \times \frac{4}{81}=\frac{100}{81}$

(iii) $\left(\frac{-2}{3}\right)^{-3} \times\left(\frac{-2}{3}\right)^{-2}=\left(\frac{3}{-2}\right)^{3} \times\left(\frac{3}{-2}\right)^{2}$

$=\frac{3^{3}}{-2^{3}} \times \frac{3^{2}}{-2^{2}}=\frac{3^{(3+2)}}{-2^{(3+2)}}=\frac{3^{5}}{-2^{5}}=\frac{-243}{32}$