Question

Verify the property: x × (y × z) = (x × y) × z by taking:

(i) $x=\frac{-7}{3},y=\frac{12}{5},z=\frac{4}{9}$

(ii) $x=0,y=\frac{-3}{5},z=\frac{-9}{4}$

(iii) $x=\frac{1}{2},y=\frac{5}{-4},z=\frac{-7}{5}$

(iv) $x=\frac{5}{7},y=\frac{-12}{13},z=\frac{-7}{18}$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}\mathrm{to}\mathrm{verify}\mathrm{that}x\times (y\times z)=(x\times y)\times z. \\ \left(\mathrm{i}\right)x=\frac{-7}{3},y=\frac{12}{5},z=\frac{4}{9} \\ \mathrm{x}\times (\mathrm{y}\times \mathrm{z})=\frac{-7}{3}\times (\frac{12}{5}\times \frac{4}{9})=\frac{-7}{3}\times \frac{16}{15}=\frac{-112}{45} \\ (x\times y)\times z=(\frac{-7}{3}\times \frac{12}{5})\times \frac{4}{9}=\frac{-28}{5}\times \frac{4}{9}=\frac{-112}{45} \\ \therefore \frac{-7}{8}\times (\frac{15}{5}\times \frac{4}{9})=(\frac{-7}{8}\times \frac{15}{5})\times \frac{4}{9} \\ \mathrm{Hence}\mathrm{verified}. \\ \left(\mathrm{ii}\right)x=0,y=\frac{-3}{5},z=\frac{-9}{4} \\ x\times (y\times z)=0\times (\frac{-3}{5}\times \frac{-9}{4})=0\times \frac{27}{20}=0 \\ (x\times y)\times z=(0\times \frac{-3}{5})\times \frac{-9}{4}=0\times \frac{-9}{4}=0 \\ \therefore 0\times (\frac{-3}{5}\times \frac{-9}{4})=(0\times \frac{-3}{5})\times \frac{-9}{4} \\ \mathrm{Hence}\mathrm{verified}. \\ \left(\mathrm{iii}\right)x=\frac{1}{2},y=\frac{5}{-4},z=\frac{-7}{4} \\ x\times (y\times z)=\frac{1}{2}\times (\frac{5}{-4}\times \frac{-7}{4})=\frac{1}{2}\times \frac{35}{16}=\frac{35}{32} \\ (x\times y)\times z=(\frac{1}{2}\times \frac{5}{-4})\times \frac{-7}{4}=\frac{5}{-8}\times \frac{-7}{4}=\frac{35}{32} \\ \therefore \frac{1}{2}\times (\frac{5}{-4}\times \frac{-7}{4})=(\frac{1}{2}\times \frac{5}{-4})\times \frac{-7}{4} \\ \mathrm{Hence}\mathrm{verified}. \\ \left(\mathrm{iv}\right)x=\frac{5}{7},y=\frac{-12}{13},z=\frac{-7}{18} \\ x\times (y\times z)=\frac{5}{7}\times (\frac{-12}{13}\times \frac{-7}{18})=\frac{5}{7}\times \frac{14}{39}=\frac{10}{39} \\ (x\times y)\times z=\frac{5}{7}\times \frac{-12}{13})\times \frac{-7}{18}=\frac{-60}{91}\times \frac{-7}{18}=\frac{10}{39} \\ \therefore \frac{5}{7}\times (\frac{-12}{13}\times \frac{-7}{18})=(\frac{5}{7}\times \frac{-12}{13})\times \frac{-7}{18} \\ \mathrm{Hence}\mathrm{verified}. \end{gathered}$$