Question

The value of x if $\sqrt[3]{39\sqrt[3]{327\sqrt[3]{381\sqrt[3]{3243}}}}=\sqrt[x]{x{3}^{x+2}}$, is

• A. $\frac{162}{17}$
• B. 9
• C. $\frac{151}{17}$
• D. 7

Answer 1

The correct option is A $\frac{162}{17}$

$\sqrt[3]{39\sqrt[3]{327\sqrt[3]{381\sqrt[3]{3243}}}}=\sqrt[x]{x{3}^{x+2}}$

$\Rightarrow \sqrt[3]{39\sqrt[3]{327\sqrt[3]{33^4\times {\left(3^5\right)}^{\frac{1}{3}}}}}=3^{\frac{{}^{x+2}}{x}}$

$\Rightarrow \sqrt[3]{39\sqrt[3]{33^3\times {\left(3^{\frac{17}{3}}\right)}^{\frac{1}{3}}}}=3^{\frac{{}^{x+2}}{x}}$

$\Rightarrow \sqrt[3]{39\sqrt[3]{33^{\frac{44}{9}}}}=3^{\frac{{}^{x+2}}{x}}$

$\Rightarrow \sqrt[3]{33^2\times 3^{\frac{44}{27}}}=3^{\frac{{}^{(x+2)}}{x}}$
$\Rightarrow {\left(3^{\frac{98}{27}}\right)}^{\frac{1}{3}}=3^{\frac{{}^{(x+2)}}{x}}$

$\Rightarrow 3^{\frac{98}{81}}=3^{\frac{{}^{x+2}}{x}}$

$\Rightarrow \frac{x+2}{x}=\frac{98}{81}$

$\Rightarrow 81x+162=98x$
$\Rightarrow 17x=162$
$\Rightarrow x=\frac{162}{17}$
Hence, the correct answer is option (a).