Verify the following :

(i) $\frac{3}{7}\times (\frac{5}{6}+\frac{12}{13})=(\frac{3}{7}\times \frac{5}{6})+(\frac{3}{7}\times \frac{12}{13})$
(ii) $\frac{-15}{4}\times (\frac{3}{7}+\frac{-12}{5})=(\frac{-15}{4}\times \frac{3}{7})+(\frac{-15}{4}\times \frac{-12}{5})$
(iii) $(\frac{-8}{3}+\frac{-13}{12})\times \frac{5}{6}=(\frac{-8}{3}\times \frac{5}{6})+(\frac{-13}{12}\times \frac{5}{6})$
(iv) $\frac{-16}{7}\times (\frac{-8}{9}+\frac{-7}{6})=(\frac{-16}{7}\times \frac{-8}{9})+(\frac{-16}{7}\times \frac{-7}{6})$

Re-arrange suitably and find the sum in each of the following :

(i)$\frac{11}{12}+\frac{-17}{3}+\frac{11}{2}+\frac{-25}{2}$

(ii)$\frac{-6}{7}+\frac{-5}{6}+\frac{-4}{9}+\frac{-15}{7}$

(iii)$\frac{3}{5}+\frac{7}{3}+\frac{9}{5}+\frac{-13}{15}+\frac{-7}{3}$

(iv) $\frac{4}{13}+\frac{-5}{8}+\frac{-8}{13}+\frac{9}{13}$

(v)$\frac{2}{3}+\frac{-4}{5}+\frac{1}{3}+\frac{2}{5}$

(vi) $\frac{1}{8}+\frac{5}{12}+\frac{2}{7}+\frac{7}{12}+\frac{9}{7}+\frac{-5}{16}$

Prove that:
(i) $\frac{1}{3+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{3}+1}=1$
(ii) $\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{8}}+\frac{1}{\sqrt{8}+\sqrt{9}}=2$

Name the property of multiplication illustrated by each of the following statements :

(i) $\frac{-15}{8}\times \frac{-12}{7}=\frac{-12}{7}\times \frac{-15}{8}$

(ii) $(\frac{-2}{3}\times \frac{7}{9})\times \frac{-9}{5}=\frac{-2}{3}\times (\frac{7}{9}\times \frac{-9}{5})$

(iii) $\frac{-3}{4}\times (\frac{-3}{4}\times \frac{-5}{6})+(\frac{-3}{4}\times \frac{7}{8})$

(iv) $\frac{-16}{9}\times 1=1\times \frac{-16}{9}=\frac{-16}{9}$

(v) $\frac{-11}{15}\times \frac{15}{-11}=\frac{15}{-11}\times \frac{-11}{15}=1$

(vi) $\frac{-7}{5}\times 0=0$

Fill in the blanks.

(i) $\left(\frac{-3}{17}\right)+\left(\frac{-12}{5}\right)=\left(\frac{-12}{5}\right)+(.....)$

(ii) $-9+\frac{-21}{8}=(......)+(-9)$

(iii) $(\frac{-8}{13}+\frac{3}{7})+\left(\frac{-13}{4}\right)=(......)+[\frac{3}{7}+(\frac{-13}{4}\left)\right]$

(iv) $-12+(\frac{7}{12}+\frac{-9}{11})=(-12+\frac{7}{12})+(.....)$

(v) $\frac{19}{-5}+(\frac{-3}{11}+\frac{-7}{8})=\{\frac{19}{-5}+(.....\left)\right\}+\frac{-7}{8}$

(vi) $\frac{-16}{7}+.....=......+\frac{-16}{7}=\frac{-16}{7}$

Verify each of the following :

(i) $\frac{3}{7}\times \frac{-5}{9}=\frac{-5}{9}\times \frac{3}{7}$

(ii) $\frac{-8}{7}\times \frac{13}{9}=\frac{13}{9}\times \frac{-8}{7}$

(iii) $\frac{-12}{5}\times \frac{7}{-36}=\frac{7}{-36}\times \frac{-12}{5}$

(iv) $-8\times \frac{-13}{12}=\frac{-13}{12}\times (-8)$