If (sqrt{3})^{5} times 9^{2} = 3^{n} times 3 | vedclass

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(B) Given equation: $(\sqrt{3})^{5} 	imes 9^{2} = 3^{n} 	imes 3 \sqrt{3}$Express all terms with base $3$:$(\sqrt{3})^{5} = (3^{1/2})^{5} = 3^{5/2}$$

Vedclass · Accepted Answer

$5$

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(B) Given equation: $(\sqrt{3})^{5} 	imes 9^{2} = 3^{n} 	imes 3 \sqrt{3}$Express all terms with base $3$:$(\sqrt{3})^{5} = (3^{1/2})^{5} = 3^{5/2}$$9^{2} = (3^{2})^{2} = 3^{4}$$3 \sqrt{3} = 3^{1} 	imes 3^{1/2} = 3^{1 + 1/2} = 3^{3/2}$Substitute these into the equation:$3^{5/2} 	imes 3^{4} = 3^{n} 	imes 3^{3/2}$Using the law of exponents $a^{m} 	imes a^{n} = a^{m+n}$:$3^{5/2 + 4} = 3^{n + 3/2}$$3^{13/2} = 3^{n + 3/2}$Equating the exponents:$13/2 = n + 3/2$$n = 13/2 - 3/2$$n = 10/2$$n = 5$

If $(\sqrt{3})^{5} \times 9^{2} = 3^{n} \times 3 \sqrt{3}$,then find the value of $n$.

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