Question

{(2½×3⅓×4¼)÷(10–⅕×5⅗)}÷{(3^4/3 × 5^-7/5)÷(4–⅗×6)}=10
Prove L.H.S = R.H.S​

Answer 1

$LHS = \Big(\frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times 4^{\frac{1}{4}}}{ 10^{\frac{-1}{5}} \times 5^{\frac{3}{5}}} \Big) \div \Big( \frac{3^{\frac{4}{3}} \times 5^{\frac{-7}{5}}}{4^{\frac{-3}{5}} \times 6}\Big)$

$= \Big(\frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times (2^{2})^{\frac{1}{4}}}{ (2\times5)^{\frac{-1}{5}} \times 5^{\frac{3}{5}}} \Big) \div \Big( \frac{3^{\frac{4}{3}} \times 5^{\frac{-7}{5}}}{(2^{2})^{\frac{-3}{5}} \times 2\times 3}\Big)$

$= \Big(\frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times 2^{\frac{2}{4}}}{ (2^{\frac{-1}{5}}\times5^{\frac{-1}{5}} \times 5^{\frac{3}{5}}} \Big) \div \Big( \frac{3^{\frac{4}{3}} \times 5^{\frac{-7}{5}}}{2^{\frac{-6}{5}} \times 2\times 3}\Big)$

$= \Big(\frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times 2^{\frac{2}{4}}}{ (2^{\frac{-1}{5}}\times5^{\frac{-1}{5}} \times 5^{\frac{3}{5}}} \Big) \times \Big( \frac{2^{\frac{-6}{5}} \times 2\times 3}{3^{\frac{4}{3}} \times 5^{\frac{-7}{5}}}\Big)$

$= 2^{\frac{1}{2}+\frac{1}{2}+\frac{1}{5}-\frac{6}{5} +1} \times 3^{\frac{1}{3} - 1 + \frac{4}{3}} \times 5^{\frac{-1}{5} + \frac{3}{5} - \frac{7}{5}}$

$= 2^{\frac{5+5+2-12+10}{10}} \times 3^{\frac{-1-3+4}{3}} \times 5^{\frac{-1+3-7}{5}}$

$= 2^{\frac{10}{10}}\times 3^{0} \times 5^{\frac{5}{5}} \\= 2\times 1 \times 5 \\= 10 \\= RHS$

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