12. Resuelve: 8 $$ \sqrt[n]{\frac{x^{n}+9^{n}}{81^{n}+x^{n}}}=\frac{1}{3} $$ $$ \begin{array}{lll} ext { A) } 27 & ext { B) } 81 & ext { C) } 16 \ ext { D) } 49 & ext { E) } 36 & \end{array} $$

Question

12. Resuelve: 8 $$ \sqrt[n]{\frac{x^{n}+9^{n}}{81^{n}+x^{n}}}=\frac{1}{3} $$ $$ \begin{array}{lll}	ext { A) } 27 & 	ext { B) } 81 & 	ext { C) } 16 \ 	ext { D) } 49 & 	ext { E) } 36 & \end{array} $$

UpStudy AI · Accepted Answer

Solve the equation $$ 8 \sqrt[n]{\frac{x^{n}+9^{n}}{81^{n}+x^{n}}}=\frac{1}{3} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$8\sqrt[n]{\frac{x^{n}+9^{n}}{81^{n}+x^{n}}}=\frac{1}{3}$$
- step1: Simplify:
  $$8\left(\frac{x^{n}+3^{2n}}{3^{4n}+x^{n}}ight)^{\frac{1}{n}}=\frac{1}{3}$$
- step2: Multiply by the reciprocal:
  $$8\left(\frac{x^{n}+3^{2n}}{3^{4n}+x^{n}}ight)^{\frac{1}{n}}	imes \frac{1}{8}=\frac{1}{3}	imes \frac{1}{8}$$
- step3: Multiply:
  $$\left(\frac{x^{n}+3^{2n}}{3^{4n}+x^{n}}ight)^{\frac{1}{n}}=\frac{1}{24}$$
- step4: Calculate:
  $$\frac{x^{n}+3^{2n}}{3^{4n}+x^{n}}=\left(\frac{1}{24}ight)^{n}$$
- step5: Rewrite the expression:
  $$\frac{x^{n}+3^{2n}}{3^{4n}+x^{n}}=\frac{1}{24^{n}}$$
- step6: Cross multiply:
  $$\left(x^{n}+3^{2n}ight)	imes 24^{n}=3^{4n}+x^{n}$$
- step7: Simplify the equation:
  $$24^{n}x^{n}+3^{2n}	imes 24^{n}=3^{4n}+x^{n}$$
- step8: Move the expression to the left side:
  $$24^{n}x^{n}+3^{2n}	imes 24^{n}-\left(3^{4n}+x^{n}ight)=0$$
- step9: Remove the parentheses:
  $$24^{n}x^{n}+3^{2n}	imes 24^{n}-3^{4n}-x^{n}=0$$
- step10: Use substitution:
  $$24^{n}t+3^{2n}	imes 24^{n}-3^{4n}-t=0$$
- step11: Collect like terms:
  $$\left(24^{n}-1ight)t+3^{2n}	imes 24^{n}-3^{4n}=0$$
- step12: Move the constant to the right side:
  $$\left(24^{n}-1ight)t=0-\left(3^{2n}	imes 24^{n}-3^{4n}ight)$$
- step13: Subtract the terms:
  $$\left(24^{n}-1ight)t=-3^{2n}	imes 24^{n}+3^{4n}$$
- step14: Divide both sides:
  $$\frac{\left(24^{n}-1ight)t}{24^{n}-1}=\frac{-3^{2n}	imes 24^{n}+3^{4n}}{24^{n}-1}$$
- step15: Divide the numbers:
  $$t=\frac{-3^{2n}	imes 24^{n}+3^{4n}}{24^{n}-1}$$
- step16: Substitute back:
  $$x^{n}=\frac{-3^{2n}	imes 24^{n}+3^{4n}}{24^{n}-1}$$
- step17: Calculate:
  $$x=\left(\frac{-3^{2n}	imes 24^{n}+3^{4n}}{24^{n}-1}ight)^{\frac{1}{n}}$$
- step18: Simplify:
  $$x=\left(\frac{-6^{3n}+3^{4n}}{24^{n}-1}ight)^{\frac{1}{n}}$$
The solution to the equation $$8 \sqrt[n]{\frac{x^{n}+9^{n}}{81^{n}+x^{n}}}=\frac{1}{3}$$ is $$x=(\frac{-6^{3n}+3^{4n}}{24^{n}-1})^{\frac{1}{n}}$$.
The solution to the equation is $$ x = 27 $$.

Resuelve: 8

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