Question

please find the answer with proper steps

Answer 1

Given : $\sqrt[3]{9\sqrt[3]{27\sqrt[3]{81\sqrt[3]{243}}} } =\sqrt[x]{3^{x+2}}$

To Find : Value of x

Solution:

$\sqrt[3]{9\sqrt[3]{27\sqrt[3]{81\sqrt[3]{243}}} } =\sqrt[x]{3^{x+2}}$

∛243 = ∛3⁵  =  $3^{\frac{5}{3}}$

∛81.$3^{\frac{5}{3}}$=∛3⁴.$3^{\frac{5}{3}}$=∛$3^{\frac{17}{3}}$=$3^{\frac{17}{9}}$

∛27.$3^{\frac{17}{9}}$=∛3³.$3^{\frac{17}{9}}$=∛$3^{\frac{44}{9}}$=$3^{\frac{44}{27}}$

∛9.$3^{\frac{44}{27}}$=∛3².$3^{\frac{44}{27}}$=∛$3^{\frac{98}{27}}$=$3^{\frac{98}{81}}$

98/81 = (x + 2)/x

=> 98x = 81x + 162

=> 17x = 162

=> x = 162/17

If

$\sqrt[3]{9\sqrt[3]{27\sqrt[3]{81\sqrt[3]{243}}} } =\sqrt[3]{3^{x+2}}$

Taking cube on both sides

=>  ${9\sqrt[3]{27\sqrt[3]{81\sqrt[3]{243}}} }=3^{x+2}$

=> $9\sqrt[3]{27\sqrt[3]{81\sqrt[3]{243}}} =3^x.3^2$

=> $9\sqrt[3]{27\sqrt[3]{81\sqrt[3]{243}}} =3^x.9$

=> $\sqrt[3]{27\sqrt[3]{81\sqrt[3]{243}}} =3^x$

Taking cube on both sides

=> $27\sqrt[3]{81\sqrt[3]{243}} =3^{3x}$

=> $3^3\sqrt[3]{81\sqrt[3]{243}} =3^{3x}$

Taking cube on both sides

=> $3^9 \cdot 81\sqrt[3]{243} =3^{9x}$

=> $3^9 \cdot 3^4\sqrt[3]{243} =3^{9x}$

=> $3^{13} \sqrt[3]{243} =3^{9x}$

Taking cube on both sides

=> $3^{39} \cdot 243 =3^{27x}$

=> $3^{39} \cdot 3^5 =3^{27x}$

=> $3^{44} =3^{27x}$

=> 27x = 44

=> x = 44/27

Learn more:

Cube root of (x+1)(y+2)(z+3)=7 Find xyz.

https://brainly.in/question/13081153

(y-3)^2=9, then cube root of the maximum value of (3x+4y)

https://brainly.in/question/11524870