(2^1/2 * 3^1/3* 4^1/4/10^-1/55^3/5) / ( 3^4/3 5^-7/5 / 4^-3/5* 6)

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Solution

$\frac{(\frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times 4^{\frac{1}{4}}}{10^{\frac{- 1}{5}} \times 5^{\frac{3}{5}}})}{(\frac{3^{\frac{4}{3}} \times 5^{\frac{- 7}{5}}}{4^{\frac{- 3}{5}} \times 6})} = \frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times 4^{\frac{1}{4}}}{10^{\frac{- 1}{5}} \times 5^{\frac{3}{5}}} \times \frac{4^{\frac{- 3}{5}} \times 6}{3^{\frac{4}{3}} \times 5^{\frac{- 7}{5}}} = \frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times {(2^{2})}^{\frac{1}{4}}}{{(2 \times 5)}^{\frac{- 1}{5}} \times 5^{\frac{3}{5}}} \times \frac{{(2^{2})}^{\frac{- 3}{5}} \times 2 \times 3}{3^{\frac{4}{3}} \times 5^{\frac{- 7}{5}}} = \frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times 2^{\frac{1}{2}}}{2^{\frac{- 1}{5}} \times 5^{\frac{- 1}{5}} \times 5^{\frac{3}{5}}} \times \frac{2^{\frac{- 6}{5}} \times 2 \times 3}{3^{\frac{4}{3}} \times 5^{\frac{- 7}{5}}} {Using the exponent law {(a^{m})}^{n} = a^{m n}} = \frac{2^{\frac{1}{2}} \times 2^{\frac{1}{2}} \times 2^{\frac{- 6}{5}} \times 2}{2^{\frac{- 1}{5}}} \times \frac{3^{\frac{1}{3}} \times 3}{3^{\frac{4}{3}}} \times \frac{1}{5^{\frac{- 1}{5}} \times 5^{\frac{3}{5}} \times 5^{\frac{- 7}{5}}} = \frac{2^{[\frac{1}{2} + \frac{1}{2} + (\frac{- 6}{5}) + 1]}}{2^{\frac{- 1}{5}}} \times \frac{3^{(\frac{1}{3} + 1)}}{3^{\frac{4}{3}}} \times \frac{1}{5^{(\frac{- 1}{5} + \frac{3}{5} + \frac{- 7}{5})}} {Using the exponent law a^{m} \times a^{n} = a^{m + n}} = \frac{2^{\frac{4}{5}}}{2^{\frac{- 1}{5}}} \times \frac{3^{\frac{4}{3}}}{3^{\frac{4}{3}}} \times \frac{1}{5^{- 1}} = 2^{\frac{4}{5} - (\frac{- 1}{5})} \times 1 \times 5 {Since; \frac{a^{m}}{a^{n}} = a^{m - n}} = 2 \times 5 = 10$

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