Question 1209806
Let's solve this problem step-by-step.

**1. Express x and y in terms of r and s**

Given:
* x⁵y¹⁷ = r  ...(1)
* x²y⁷ = s   ...(2)

We want to find x and y in the form:
* x = r^a / s^b
* y = s^c / r^d

**2. Solve for x and y**

We need to eliminate one of the variables to express the other in terms of r and s.

* **Eliminate y:**
    * Raise equation (2) to the power of 17: (x²y⁷)¹⁷ = s¹⁷  => x³⁴y¹¹⁹ = s¹⁷
    * Raise equation (1) to the power of 7: (x⁵y¹⁷)⁷ = r⁷ => x³⁵y¹¹⁹ = r⁷
    * Divide the second equation by the first: (x³⁵y¹¹⁹) / (x³⁴y¹¹⁹) = r⁷ / s¹⁷
    * x = r⁷ / s¹⁷

* **Eliminate x:**
    * Raise equation (2) to the power of 5: (x²y⁷)⁵ = s⁵ => x¹⁰y³⁵ = s⁵
    * Raise equation (1) to the power of 2: (x⁵y¹⁷)² = r² => x¹⁰y³⁴ = r²
    * Divide the first equation by the second: (x¹⁰y³⁵) / (x¹⁰y³⁴) = s⁵ / r²
    * y = s⁵ / r²

**3. Compare with the Given Forms**

We have:
* x = r⁷ / s¹⁷ = r⁷s⁻¹⁷
* y = s⁵ / r² = s⁵r⁻²

Comparing with the given forms:
* x = r^a / s^b = r^a s^-b
* y = s^c / r^d = s^c r^-d

We can see:
* a = 7
* b = 17
* c = 5
* d = 2

**4. Calculate a + b + c + d**

* a + b + c + d = 7 + 17 + 5 + 2 = 31

**Therefore, a + b + c + d = 31**