Question

Pls solve this i'll thank you and the first to give right ans I'll mark them brainliets

Answer 1

Question

• $\sf If~ 3^x = 2^y , then\ the \ value\ of\ \: \bf\sqrt[y]{ {243}^{x} }$

Solution

$\: \: \: \: \: : \: \: \implies \: \sf\sqrt[y]{ {243}^{x}}$

• Convert 243 into its prime factors

$\: \: \: \: \: : \: \: \implies \: \sf\sqrt[y]{ (3 \times 3 \times 3 \times 3 \times 3)^{x}}$

• $\\\gray{\sf {a}^{m} \times {a}^{n} = {a}^{m + n}}$

$\: \: \: \: \: : \: \: \implies \: \sf\sqrt[y]{ ( {3}^{1 + 1 + 1 + 1 + 1} )^{x}}$

$\: \: \: \: \: : \: \: \implies \: \sf\sqrt[y]{ ( {3}^{5} )^{x}}$

$\: \: \: \: \: : \: \: \implies \: \sf\sqrt[y]{ ( 3)^{ x \times 5}}$

$\: \: \: \: \: : \: \: \implies \: \sf\sqrt[y]{ ( {3}^{x})^{ 5}}$

• $\sf~~ putting \: 3^x = 2^y \: that \: is \: given$

$\: \: \: \: \: : \: \: \implies \: \sf\sqrt[y]{ ( {2}^{y} )^{ 5}}$

$\: \: \: \: \: : \: \: \implies \: \sf\sqrt[y]{ {2}^{ 5y}}$

${~~~~~\bigstar~~~~\gray{\sf x^{\frac{m}{n} }=\sqrt[n]{x^m}\sf = (\sqrt[n]{x})^m}}$

$\: \: \: \: \: : \: \: \implies \: \sf {2}^{ \frac{5\cancel{y}}{\cancel{y}} }$

$\: \: \: \: \: : \: \: \implies \: \sf {2}^{ 5 }$

$\: \: \: \: \: : \: \: \implies \: \sf 32$

$\: \: \bold{\sqrt[y]{ {243}^{x}} = 32}$

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$\small\begin{gathered}\begin{gathered}\\\\\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \bigstar \: \underline{\bf{}}\\ {\boxed{\begin{array}{c | c} \frac{ \: ~~~~~~~~~~\: \: \: \: \:\sf Laws \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }{ } &\frac{ \: ~~~~~~~~~~ \: \: \: \: \:\sf Example \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }{ }\\ \sf \bigstar{a}^{m} \times {a}^{n} = {a}^{m + n} & \sf {a}^{2} \times {a}^{3} = {a}^{2 + 3} = {a}^{6} \\ \\ \sf \bigstar{a}^{m} \div {a}^{n} = {a}^{m - n}& \sf {a}^{3} \div {a}^{2} = {a}^{3 - 2} = {a}^{1} \\ \\ \sf{\bigstar \: \: \: \: \: \: ( {a}^{m} ) ^{n} = {a}^{mn} } & \sf( {a}^{2} ) ^{3} = {a}^{2 \times 3} = {a}^{6} \\ \\ {\bigstar\sf a {}^{m} \times {n}^{m} = (ab) ^{m} } &\sf a {}^{2} \times {b}^{2} = (ab) ^{2}\\ \\ \sf\bigstar \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: {a}^{0} = 1& \sf {2}^{0} = 1 \: \: \: \: \\ \\ \sf \bigstar \: \: \: \: {\dfrac{ {a}^{m} }{ {b}^{m} }= \left( \dfrac{a}{b} \right) ^{m} }& \sf{\dfrac{ {a}^{2} }{ {b}^{2} }= \left( \dfrac{a}{b} \right) ^{2} }\\\\\bigstar~~~~~~~ \sf x^{\frac{m}{n} }=\sqrt[n]{x^m}\sf = (\sqrt[n]{x})^m & \sf x^{\frac{2}{3} }=\sqrt[3]{x^2} = (\sqrt[n]{x})^m\\ \\\\ \end{array}}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

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