Question

What will come in place of question mark (?) in the following question?

\(\frac{{{{216}^{\frac{2}{3}}} \times {{45}^2} \times 405 \times {6^3}}}{{256 \times {{135}^2} \times {{27}^{\frac{7}{3}}}}} = {\rm{\;}}?\)

1. \(\frac{{25}}{8}\)
2. \(\frac{{5}}{8}\)
3. \(\frac{{35}}{8}\)
4. \(\frac{{15}}{16}\)
5. None of the above

Answer 1

Correct Answer - Option 2 : \(\frac{{5}}{8}\)

Given expression

\(\begin{array}{l} \frac{{{{216}^{\frac{2}{3}}}\; \times \;{{45}^2}\; \times 405\; \times {6^3}}}{{256\; \times {{135}^2}\; \times {{27}^{\frac{7}{3}}}}}\\ \Rightarrow \frac{{{{\left( {{6^3}} \right)}^{\frac{2}{3}}}\; \times \;{{\left( {5\; \times \;{3^2}} \right)}^2}\; \times \;5\; \times 81\; \times {6^3}}}{{{2^8}\; \times {{\left( {5\; \times \;{3^3}} \right)}^2}\; \times {{\left( {{3^3}} \right)}^{\frac{7}{3}}}}}\\ \Rightarrow \frac{{{6^2}\; \times \;{5^2}\; \times \;{3^4}\; \times \;5\; \times \;{3^4}\; \times {6^3}}}{{{2^8}\; \times \;{5^2}\; \times \;{3^6}\; \times \;{3^7}}}\\ \Rightarrow \frac{{{6^{2 + 3}}\; \times \;{5^{2 + 1}}\; \times \;{3^{4 + 4}}}}{{{2^8}\; \times \;{5^2}\; \times \;{3^{6 + 7}}}}\\ \Rightarrow \frac{{{{\left( {2\; \times 3} \right)}^5}\; \times \;{5^3}\; \times \;{3^8}}}{{{2^8}\; \times \;{5^2}\; \times \;{3^{13}}}}\\ \Rightarrow \frac{{{2^5}\; \times \;{5^3}\; \times \;{3^{8 + 5}}}}{{{2^8}\; \times \;{5^2}\; \times \;{3^{13}}}}\\ \Rightarrow \frac{{{5^{3 - 2}}\; \times \;{3^{13 - 13}}}}{{{2^{8 - 5}}}}\\ \Rightarrow \frac{{5 }}{{{2^3}}} = 5/8 \end{array}\)