If $$ x $$ is a positive real number such that
$$ x^8 + \left(\frac{1}{x}\right)^8 = 47 $$,
then the value of
$$ x^9 + \left(\frac{1}{x}\right)^9 $$ is:

Question

If $$ x $$ is a positive real number such that  
$$ x^8 + \left(\frac{1}{x}\right)^8 = 47 $$,  
then the value of  
$$ x^9 + \left(\frac{1}{x}\right)^9 $$ is:

Accepted Answer

We are given that:  
$$ x^8 + \left(\frac{1}{x}\right)^8 = 47. $$

1. **Step 1: Express higher powers using lower ones**  
   Using the property of symmetric expressions, calculate:  
   $$
   \left(x^4 + \frac{1}{x^4}\right)^2 = x^8 + \frac{1}{x^8} + 2.
   $$
   Substituting the given value:  
   $$
   x^8 + \frac{1}{x^8} = 47 \implies x^4 + \frac{1}{x^4} = \sqrt{49} = 7.
   $$

2. **Step 2: Calculate lower powers recursively**  
   Similarly, use:  
   $$
   \left(x^2 + \frac{1}{x^2}\right)^2 = x^4 + \frac{1}{x^4} + 2.
   $$
   Substituting:  
   $$
   x^4 + \frac{1}{x^4} = 7 \implies x^2 + \frac{1}{x^2} = \sqrt{9} = 3.
   $$
   Next, compute:  
   $$
   x + \frac{1}{x} = \sqrt{5}.
   $$

3. **Step 3: Calculate higher powers**  
   Using the binomial expansion for symmetric expressions:  
   $$
   x^3 + \frac{1}{x^3} = (x + \frac{1}{x})\left(x^2 + \frac{1}{x^2}\right) - (x + \frac{1}{x}) = 3\sqrt{5}.
   $$
   Similarly, calculate:  
   $$
   x^9 + \frac{1}{x^9} = (x^3 + \frac{1}{x^3})\left(x^6 + \frac{1}{x^6}\right) - (x^3 + \frac{1}{x^3})(x^3 + \frac{1}{x^3}).
   $$
   Substituting values step-by-step yields:  
   $$
   x^9 + \frac{1}{x^9} = 34\sqrt{5}.
   $$

Thus, the correct answer is **$$ 34\sqrt{5} $$**.

Answer

Answer

Answer