Question

evaluate
[(12/25)² × (15/2) ³ ] ÷ (9/2) ² solution

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Answer 1

Solution :

$\sf \bigg[ \bigg( \dfrac{12}{25} \bigg)^2 \times \bigg( \dfrac{15}{2} \bigg)^3\bigg] \div \bigg( \dfrac{9}{2} \bigg)^2$

$\sf \implies\: \bigg[ \bigg( \dfrac{2 \times 2 \times 3}{5 \times 5} \bigg)^2 \times \bigg( \dfrac{3 \times 5}{2} \bigg)^3\bigg] \div \bigg( \dfrac{3 \times 3}{2} \bigg)^2$

$\sf \implies\:\bigg[ \bigg( \dfrac{ {2}^{2} \times 3}{ {5}^{2} } \bigg)^2 \times \bigg( \dfrac{3 \times 5}{2} \bigg)^3\bigg] \div \bigg( \dfrac{ {3}^{2} }{2} \bigg)^2$

Using identity

$\sf \bullet \: \bigg( \dfrac{a}{b} \bigg)^2 = \dfrac{ {a}^{2} }{ {b}^{2} }$

$\sf \bullet \: { ({a}^{b}) }^{c} \: = \: {a}^{(b \times c)}$

$\sf \bullet \: {a \times b}^{c} \: = \: {a}^{c} \: \times \: {b}^{c}$

We get,

$\sf \implies \: \bigg[ \: \bigg( \dfrac{ ({2}^{2}) {}^{2} \times {3}^{2} }{ ({5}^{2} ){}^{2} } \bigg) \times \bigg( \dfrac{ {3}^{3} \times {5}^{3} }{ {2}^{3} } \bigg)\bigg] \div \bigg( \dfrac{( {3}^{2}) {}^{2} }{ {2}^{2} } \bigg)$

$\sf \implies \: \bigg[ \: \dfrac{( {2}^{2 \times 2}) \times {3}^{2} }{ ({5}^{2 \times 2} ) } \times \dfrac{ {3}^{3} \times {5}^{3} }{ {2}^{3} } \bigg] \div \bigg( \dfrac{( {3}^{2 \times 2}) }{ {2}^{2} } \bigg)$

$\sf \implies \: \bigg[ \: \dfrac{ {2}^{4} \times {3}^{2} }{ ({5}^{4} ) } \times \dfrac{ {3}^{3} \times {5}^{3} }{ {2}^{3} } \bigg] \div \bigg( \dfrac{( {3}^{4}) }{ {2}^{2} } \bigg)$

$\sf \implies \: \bigg[ \: \dfrac{ {2}^{4} \times {3}^{2} \times \: {3}^{3} \times {5}^{3} }{ {5}^{4} \times {2}^{3} } \bigg] \: \div \bigg( \dfrac{( {3}^{4}) }{ {2}^{2} } \bigg)$

Using identity,

$\sf \bullet \: {a}^{m} \times {a}^{n } \: = \: {a}^{(m + n)}$

$\sf \bullet \: \dfrac{ {a}^{m} }{ {a}^{n} } \: = \: \: {a}^{(m - n)}$

We get,

$\sf \implies \: \bigg[ \: \dfrac{ {2}^{(4 - 3)} \times {3}^{(2 + 3)} }{ {5}^{(4 - 3)} } \bigg] \: \div \bigg( \dfrac{( {3}^{4}) }{ {2}^{2} } \bigg)$

$\sf \implies \: \bigg[ \: \dfrac{ {2}^{(1)} \times {3}^{(5)} }{ {5}^{(1)} } \bigg] \: \div \bigg( \dfrac{( {3}^{4}) }{ {2}^{2} } \bigg)$

$\sf \implies \: \: \dfrac{ {2}^{(1)} \times {3}^{(5)} }{ {5}^{(1)} } \: \div \dfrac{( {3}^{4}) }{ {2}^{2} }$

$\sf \implies \: \: \dfrac{ {2}^{(1)} \times {3}^{(5)} }{ {5}^{(1)} } \: \times \dfrac{ {2}^{2} }{ {3}^{4} }$

$\sf \implies \: \: \dfrac{ {2}^{(1 + 2)} \times {3}^{(5 - 4)} }{ {5}^{(1)} } \:$

$\sf \implies \: \: \dfrac{ {2}^{(3)} \times {3}^{(1)} }{ {5} } \:$

$\sf \implies \: \: \dfrac{ {8} \times {3} }{ {5} } \:$

$\sf \implies \: \: \dfrac{ 24 }{ {5} } \:$

Answer : 24/5