\( 1.3 \quad \frac { ( x + 1 ) ^ { 3 } ( x - 1 ) ^ { - 3 } } { ( x + 1 ) ^ { 2 } ( x - 1 ) ^ { - 2 } } \)

We start with the expression \[ \frac{(x+1)^3 (x-1)^{-3}}{(x+1)^2 (x-1)^{-2}}. \] **Step 1.** Simplify the factors with the base \(x+1\) by subtracting the exponent in the denominator from the numerator: \[ (x+1)^{3-2} = (x+1)^1 = x+1. \] **Step 2.** Simplify the factors with the base \(x-1\) by subtracting the exponent in the denominator from the numerator: \[ (x-1)^{-3-(-2)} = (x-1)^{-3+2} = (x-1)^{-1}. \] Recall that a negative exponent means taking the reciprocal: \[ (x-1)^{-1} = \frac{1}{x-1}. \] **Step 3.** Combine the simplified parts: \[ (x+1) \cdot \frac{1}{x-1} = \frac{x+1}{x-1}. \] Thus, the simplified expression is \[ \frac{x+1}{x-1}. \]