Simplify and write the answer in the exponential form. (i) \({\left(\frac{{3}^{{7}}}{{3}^{{2}}}\right)}\times{3}^{{5}}\) (ii) \({2}^{{3}}\times{2}^{{2}}\times{5}^{{5}}\)

(i) \(\frac{{{3}^{{7}}}}{{{3}^{{2}}}}\cdot{3}^{{5}}=\frac{{{3}^{{7}}\cdot{3}^{{5}}}}{{{3}^{{2}}}}\)

\(=\frac{{3}^{{{7}+{5}}}}{{2}^{{2}}}\)

\(=\frac{{3}^{{12}}}{{3}^{{2}}}\)

\(={3}^{{{12}-{2}}}={3}^{{10}}\)

(ii) \({2}^{{3}}\cdot{2}^{{2}}\cdot{5}^{{5}}\)

\(={2}^{{5}}\cdot{5}^{{5}}\)

\(={\left({2}\cdot{5}\right)}^{{5}}={10}^{{5}}\)

(iii) \({\left({6}^{{2}}\cdot{6}^{{4}}\right)}\div{6}^{{3}}\)

\(={6}^{{{2}+{4}-{3}}}={6}^{{3}}\)

(iv) \({\left[{\left({2}\right)}^{{3}}\cdot{3}^{{6}}\right]}\cdot{5}^{{6}}\)

\(={2}^{{{2}\cdot{3}}}\cdot{3}^{{6}}\cdot{5}^{{6}}\)

\(={\left({2}\cdot{3}\cdot{6}\right)}^{{6}}\)

\(={30}^{{6}}\)

answer

\(=\frac{{3}^{{{7}+{5}}}}{{2}^{{2}}}\)

\(=\frac{{3}^{{12}}}{{3}^{{2}}}\)

\(={3}^{{{12}-{2}}}={3}^{{10}}\)

(ii) \({2}^{{3}}\cdot{2}^{{2}}\cdot{5}^{{5}}\)

\(={2}^{{5}}\cdot{5}^{{5}}\)

\(={\left({2}\cdot{5}\right)}^{{5}}={10}^{{5}}\)

(iii) \({\left({6}^{{2}}\cdot{6}^{{4}}\right)}\div{6}^{{3}}\)

\(={6}^{{{2}+{4}-{3}}}={6}^{{3}}\)

(iv) \({\left[{\left({2}\right)}^{{3}}\cdot{3}^{{6}}\right]}\cdot{5}^{{6}}\)

\(={2}^{{{2}\cdot{3}}}\cdot{3}^{{6}}\cdot{5}^{{6}}\)

\(={\left({2}\cdot{3}\cdot{6}\right)}^{{6}}\)

\(={30}^{{6}}\)

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