question_answer 9)

Using laws of exponents, simplify and write the answer in exponential form: (i) \[{{3}^{2}}\times {{3}^{4}}\times {{3}^{8}}\]                   (ii) \[{{6}^{15}}\div {{6}^{10}}\]                  (iii) \[{{a}^{3}}\times {{a}^{2}}\]                        (iv) \[{{7}^{x}}\times {{7}^{2}}\] (v) \[{{({{5}^{2}})}^{3}}\div {{5}^{3}}\]                    (vi) \[{{2}^{5}}\times {{5}^{5}}\]                                (vii) \[{{a}^{4}}\times {{b}^{4}}\]                       (viii) \[{{({{3}^{4}})}^{3}}\] (ix) \[({{2}^{20}}\div {{2}^{15}})\times {{2}^{3}}\]

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)  \[{{\mathbf{8}}^{\mathbf{t}}}\mathbf{\div }{{\mathbf{8}}^{\mathbf{2}}}\]                 \[{{8}^{t}}\div {{8}^{2}}={{8}^{t-2}}\]                      \[\left| {{a}^{m}}\div {{a}^{n}}={{a}^{m-n}} \right.\]