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The given expression is:

\[
C = \sqrt[60]{\frac{1 + x + x^2 + x^3 + \cdots + x^{60}}{1 + x^{-1} + x^{-2} + x^{-3} + \cdots + x^{-60}}}, \quad (x \in \mathbb{R}^{+})
\]

### Solution:

Let's analyze and simplify the expression step by step.

#### Numerator:
The numerator is the sum of powers of \(x\) from \(x^0\) to \(x^{60}\):
\[
1 + x + x^2 + x^3 + \cdots + x^{60}.
\]
This is a geometric series, and its sum can be written as:
\[
S_{\text{num}} = \frac{1 - x^{61}}{1 - x}, \quad \text{for } x \neq 1.
\]

#### Denominator:
The denominator is the sum of negative powers of \(x\):
\[
1 + x^{-1} + x^{-2} + x^{-3} + \cdots + x^{-60}.
\]
This is also a geometric series with the first term \(1\) and common ratio \(x^{-1}\), and its sum is:
\[
S_{\text{den}} = \frac{1 - x^{-61}}{1 - x^{-1}}, \quad \text{for } x \neq 1.
\]
Since \(x^{-1} = \frac{1}{x}\), the denominator becomes:
\[
S_{\text{den}} = \frac{1 - x^{-61}}{\frac{1}{x} - 1} = \frac{x(1 - x^{-61})}{1 - x}.
\]

#### Combining:
Now, the expression for \(C\) becomes:
\[
C = \sqrt[60]{\frac{\frac{1 - x^{61}}{1 - x}}{\frac{x(1 - x^{-61})}{1 - x}}}
= \sqrt[60]{\frac{1 - x^{61}}{x(1 - x^{-61})}}.
\]

Multiplying the terms inside the square root:
\[
C = \sqrt[60]{\frac{(1 - x^{61})x^{61}}{x}} = \sqrt[60]{x^{60}} = x.
\]

Thus, the solution is:
\[
C = x.
\]

Would you like more details or have any questions about the steps?

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Here are 5 related questions for deeper understanding:
1. What is a geometric series, and how do you find its sum?
2. What is the role of the exponent \(60\) in this problem, and how does it affect the outcome?
3. How would the result change if the series involved different powers or limits?
4. Why does \(x\) need to belong to \( \mathbb{R}^{+} \) (positive real numbers)?
5. What happens to the expression if \(x = 1\)? How would we approach that case?

**Tip**: When working with geometric series, always check if the common ratio is greater than or less than 1 to determine if it converges.

Solve the equation for C: C = (root of degree 60) of ((1 + x + x^2 + x^3 + ... + x^60) / (1 + x^-1 + x^-2 + x^-3 + ... + x^-60)), where x belongs to positive real numbers.

Learn how to solve a geometric series problem involving powers of x from -60 to +60. This problem demonstrates the application of geometric series sums, exponents, and expression simplification, suitable for undergraduate or advanced high school students.

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