Answer:
(i) \[{{\mathbf{3}}^{\mathbf{2}}}\mathbf{\times }{{\mathbf{3}}^{\mathbf{4}}}\mathbf{\times }{{\mathbf{3}}^{\mathbf{8}}}\] \[{{3}^{2}}\times {{3}^{4}}\times {{3}^{8}}={{3}^{2+4}}\times {{3}^{8}}\] \[\left| {{a}^{m}}\times {{a}^{n}} \right.={{a}^{m+n}}\] \[={{3}^{6}}\times {{3}^{8}}={{3}^{6+8}}={{3}^{14}}\] \[\left| {{a}^{m}}\times {{a}^{n}} \right.={{a}^{m+n}}\] (ii) \[{{\mathbf{6}}^{\mathbf{15}}}\mathbf{\div }{{\mathbf{6}}^{\mathbf{10}}}\] \[{{6}^{15}}\div {{6}^{10}}={{6}^{15-10}}\] \[={{6}^{5}}\] \[\left| {{a}^{m}}\div {{a}^{n}} \right.={{a}^{m-n}}\] (iii) \[{{\mathbf{a}}^{\mathbf{3}}}\mathbf{\times }{{\mathbf{a}}^{\mathbf{2}}}\] \[{{a}^{3}}\times {{a}^{2}}={{a}^{3+2}}\] \[\left| {{a}^{m}}\times {{a}^{n}} \right.={{a}^{m+n}}\] \[={{a}^{5}}\] (iv) \[{{\mathbf{7}}^{\mathbf{x}}}\mathbf{\times }{{\mathbf{7}}^{\mathbf{2}}}\] \[{{7}^{x}}\times {{7}^{2}}={{7}^{x+2}}\] \[\left| {{a}^{m}}\times {{a}^{n}} \right.={{a}^{m+n}}\] (v) \[{{\mathbf{(}{{\mathbf{5}}^{\mathbf{2}}}\mathbf{)}}^{\mathbf{3}}}\mathbf{\div }{{\mathbf{5}}^{\mathbf{3}}}\] \[{{({{5}^{2}})}^{3}}\div {{5}^{3}}={{5}^{2\times 3}},{{5}^{3}}\] \[{{\left| ({{a}^{m}}) \right.}^{n}}={{a}^{mn}}\] \[={{5}^{6}},{{5}^{3}}={{5}^{6-3}}\] \[\left| {{a}^{m}}\div {{a}^{n}}={{a}^{m-n}} \right.\] \[={{5}^{3}}\] (vi) \[{{\mathbf{2}}^{\mathbf{5}}}\mathbf{\times }{{\mathbf{5}}^{\mathbf{5}}}\] \[{{2}^{5}}\times {{5}^{5}}={{(2\times 5)}^{5}}\] \[\left| {{a}^{m}}\times {{b}^{m}}={{(ab)}^{m}} \right.\] \[={{10}^{5}}\] (vii) \[{{\mathbf{a}}^{\mathbf{4}}}\mathbf{\times }{{\mathbf{b}}^{\mathbf{4}}}\] \[{{a}^{4}}\times {{b}^{4}}={{(ab)}^{4}}\] \[\left| {{a}^{m}}\times {{b}^{m}}={{(ab)}^{m}} \right.\] (viii) \[{{\mathbf{(}{{\mathbf{3}}^{\mathbf{4}}}\mathbf{)}}^{\mathbf{3}}}\] \[{{({{3}^{4}})}^{3}}={{3}^{4}}^{\times 3}\] \[\left| {{({{a}^{m}})}^{n}} \right.={{a}^{mn}}\] \[={{3}^{12}}\] (ix) \[\mathbf{(}{{\mathbf{2}}^{\mathbf{20}}}\mathbf{\div }{{\mathbf{2}}^{\mathbf{15}}}\mathbf{)\times }{{\mathbf{2}}^{\mathbf{3}}}\] \[({{2}^{20}}\div {{2}^{15}})\times {{2}^{3}}={{2}^{20-15}}\times {{2}^{3}}\] \[\left| {{a}^{m}}\div {{a}^{n}}={{a}^{m-n}} \right.\] \[={{2}^{5}}\times {{2}^{3}}\] \[={{2}^{5+3}}\] \[\left| {{a}^{m}}\div {{a}^{n}}={{a}^{m+n}} \right.\] \[={{2}^{8}}\] (x) \[{{\mathbf{8}}^{\mathbf{t}}}\mathbf{\div }{{\mathbf{8}}^{\mathbf{2}}}\] \[{{8}^{t}}\div {{8}^{2}}={{8}^{t-2}}\] \[\left| {{a}^{m}}\div {{a}^{n}}={{a}^{m-n}} \right.\]
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