Question

3^(x^2) :3^x=9:1, find x​

Answer 1

$Given \: 3^{(x^{2})} : 3^{x} = 9 : 1$

$\implies 3^{(x^{2})} \times 1 = 3^{x} \times 9$

$\blue{ ( Product \: of \: extrems = Product \: of \: means ) }$

$\implies 3^{(x^{2})} = 3^{x} \times 3^{2}$

$\implies 3^{(x^{2})} = 3^{x+2}$

/* By Exponential Law */

$\boxed{ \pink{ x^{m} \times x^{n} = x^{m+n} }}$

$\implies x^{2} = x + 2$

$\implies x^{2} -x - 2 = 0$

$\implies x^{2} -2x+1x - 2 = 0$

$\implies x(x-2) +1(x-2) = 0$

$\implies (x-2)(x+1) = 0$

$\implies x-2 = 0 \: Or \: x+1 = 0$

$\implies x = 2 \: Or \: x = -1$

Therefore.,

$\green { x = 2 \: Or \: x = -1}$

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Answer 2

${\tt{\purple{\underline{\underline{\huge{Answer:}}}}}}$

${\pink{\boxed{Given}}}$

$\sf\implies \: 3^{(x^{2})} : 3^{x} = 9 : 1$

$\implies 3^{(x^{2})} \times 1 = 3^{x} \times 9$

$( Product \: of \: extrems = Product \: of \: means$

$\implies 3^{(x^{2})} = 3^{x} \times 3^{2}$

$\implies 3^{(x^{2})} = 3^{x+2}$

$\boxed{ x^{m} \times x^{n} = x^{m+n} }$

$\implies x^{2} = x + 2$

$\implies x^{2} -x - 2 = 0$

$\implies x^{2} -2x+1x - 2 = 0$

$\implies x(x-2) +1(x-2) = 0$

$\implies (x-2)(x+1) = 0$

$\implies x-2 = 0 \: Or \: x+1 = 0$

$\implies x = 2 \: Or \: x = -1$

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