Question

simplify {2^(m+1)*3^(2m-n)*5^(m+n)*6^n} /6^m*10^(n+2)*15^m​

Answer 1

Question :-

Simplify :-

• $\sf \frac{{2}^{(m + 1)} \times {3}^{(2m - n)} \times {5}^{(m + n)} \times {6}^{n} }{ {6}^{m} \times {10}^{(n + 2)} \times {15}^{m} }$

Solution :-

$\sf \frac{{2}^{(m + 1)} \times {3}^{(2m - n)} \times {5}^{(m + n)} \times {6}^{n} }{ {6}^{m} \times {10}^{(n + 2)} \times {15}^{m} } \\ \\ \implies \sf \: \sf \frac{{2}^{(m + 1)} \times {3}^{(2m - n)} \times {5}^{(m + n)} \times {6}^{n} }{ {6}^{m} \times {(5 \times 2)}^{(n + 2)} \times {(5 \times 3)}^{m} } \\ \\ \implies \sf \: \sf \frac{{2}^{(m + 1)} \times {3}^{(2m - n)} \times {5}^{(m + n)} \times {6}^{n} }{ {6}^{m} \times {5}^{(n + 2)} \times {2}^{(n + 2)} \times {5}^{m} \times {3}^{m} } \\ \\ \implies \sf \: \frac{ {2}^{(m + 1)} }{ {2}^{(n + 2)} } \times \frac{ {3}^{(2m - n)} }{ {3}^{m} } \times \frac{ {5}^{(m + n)} }{ {5}^{(n + 2)} \times {5}^{m} } \times \frac{ {6}^{n} }{ {6}^{m} } \\ \\ \implies \sf \: {2}^{(m + 1)} \times {2}^{ - (n + 2)} \times {3}^{(2m - n)} \times {3}^{ - m} \times {5}^{(m + n)} \times {5}^{ - (n + 2)} \times {5}^{ - m} \times {6}^{n} \times {6}^{ - n} \\ \\ \implies \sf \: {2}^{(m + \cancel{n }- \cancel{n} - 2)} \times {3}^{(2m - n - m)} \times {5}^{(\cancel{m} + \cancel{n} - \cancel{n }- 2 - \cancel{m})} \times {6}^{n - m} \\ \\ \implies \sf \: {2}^{(m - 2)} \times {3}^{(m - n)} \times {5}^{ - 2} \times {6}^{(n - m)} \\ \\ \implies \sf \: {2}^{(m - 2)} \times {3}^{(m - n)} \times {5}^{ - 2} \times {(2 \times 3)}^{(n - m)} \\ \\ \implies \sf \: {2}^{(m - 2)} \times {3}^{(m - n)} \times {5}^{ - 2} \times {2}^{(n - m)} \times {3}^{(n - m)} \\ \\ \implies \sf \: {2}^{( \cancel{m} - 2 + n - \cancel{m})} \times {3}^{(\cancel{m} - \cancel{n} + \cancel{n} - \cancel{m})} \times {5}^{ - 2} \\ \\ \implies \sf \: {2}^{(n - 2)} \times {3}^{0} \times {5}^{ - 2} \\ \\ \implies \sf \: {2}^{(n - 2)} \times 1 \times \frac{1}{ {5}^{2} } \\ \\ \implies \sf \: \frac{ {2}^{(n - 2)} }{ {5}^{2} }$

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Hence,

$\boxed{ \sf \frac{{2}^{(m + 1)} \times {3}^{(2m - n)} \times {5}^{(m + n)} \times {6}^{n} }{ {6}^{m} \times {10}^{(n + 2)} \times {15}^{m} } = \frac{ {2}^{(n - 2)} }{ {5}^{2} } }$