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**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**
