Answer:
S = [tex]\frac{36}{25}[/tex]
Step-by-step explanation:
Given expression is,
S = [tex]\sqrt[3]{[(\frac{3}{2})^{-2}+6^{-3}+(\frac{3}{2})^{-3}-6^{-1}}]^{-2}[/tex]
S = [tex]\sqrt[3]{[(\frac{2}{3})^{2}+(\frac{1}{6})^{3}+(\frac{2}{3})^{3}-(\frac{1}{6})}]^{-2}[/tex]
= [tex]\sqrt[3]{[\frac{4}{9}+\frac{1}{216}+\frac{8}{27}-\frac{1}{6}}]^{-2}[/tex]
= [tex]\sqrt[3]{(\frac{96+1+64-36}{216})^{-2}}[/tex]
= [tex]\sqrt[3]{(\frac{125}{216})^{-2}}[/tex]
= [tex]\sqrt[3]{(\frac{216}{125})^{2}}[/tex]
= [tex]\sqrt[3]{(\frac{216}{125})\times (\frac{216}{125})}[/tex]
= [tex]\sqrt[3]{(\frac{6\times 6\times 6}{5\times 5\times 5})\times (\frac{6\times 6\times 6}{5\times 5\times 5})}[/tex]
= [tex]\sqrt[3]{(\frac{6}{5})^3\times (\frac{6}{5})^3}[/tex]
= [tex]\frac{6}{5}\times \frac{6}{5}[/tex]
= [tex]\frac{36}{25}[/tex]
