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The given problem is asking for the value of \( k \) in the expansion of the expression:

\[
\left( kx + \frac{1}{x} \right)^5 + \left( 1 - \frac{2}{x} \right)^8
\]
where the coefficient of \( \frac{1}{x} \) is 74.

### Step-by-step approach:

We need to expand both binomials and find the terms that contain \( \frac{1}{x} \), then equate their coefficients to 74.

1. **Expand \( \left( kx + \frac{1}{x} \right)^5 \):**

Using the binomial theorem:
\[
\left( kx + \frac{1}{x} \right)^5 = \sum_{r=0}^{5} \binom{5}{r} (kx)^{5-r} \left( \frac{1}{x} \right)^r
\]
Simplify each term:
\[
\binom{5}{r} k^{5-r} x^{5-2r}
\]
For the coefficient of \( \frac{1}{x} \), we need \( 5 - 2r = -1 \), which gives \( r = 3 \).

So, the term containing \( \frac{1}{x} \) is:
\[
\binom{5}{3} k^{2} = \frac{5!}{3!2!} k^{2} = 10k^2
\]

2. **Expand \( \left( 1 - \frac{2}{x} \right)^8 \):**

Using the binomial theorem:
\[
\left( 1 - \frac{2}{x} \right)^8 = \sum_{r=0}^{8} \binom{8}{r} 1^{8-r} \left( -\frac{2}{x} \right)^r
\]
Simplify each term:
\[
\binom{8}{r} (-2)^r \left( \frac{1}{x} \right)^r
\]
For the coefficient of \( \frac{1}{x} \), we need \( r = 1 \).

So, the term containing \( \frac{1}{x} \) is:
\[
\binom{8}{1} (-2)^1 = 8(-2) = -16
\]

3. **Combine the terms:**

The total coefficient of \( \frac{1}{x} \) is:
\[
10k^2 - 16 = 74
\]
Solving this equation:
\[
10k^2 = 74 + 16 = 90
\]
\[
k^2 = \frac{90}{10} = 9
\]
\[
k = 3
\]

### Final answer:
The value of the positive constant \( k \) is \( \boxed{3} \).

Would you like more details on the steps, or do you have any other questions?

### Related questions:
1. How does the binomial theorem work for negative powers or fractional exponents?
2. Can you explain the role of the binomial coefficients in the expansion?
3. What would the solution look like if the exponent values were different in this problem?
4. How do we handle complex binomial expansions involving both fractions and variable powers?
5. What are the general applications of binomial expansions in physics or other sciences?

### Tip:
When solving binomial expansion problems, it’s helpful to identify the specific terms you need by focusing on powers of \( x \) in the expansion.

The coefficient of 1/x in the expansion of (kx + 1/x)^5 + (1 - 2/x)^8 is 74. Find the value of the positive constant k.

This problem involves finding the coefficient of 1/x in the binomial expansions of two expressions, and solving for the constant k. It uses the binomial theorem and algebraic manipulation to match coefficients.

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