(с+3/с-3 - с/с+3) • с-3/с+1

To simplify the expression \(\left(\frac{c+3}{c-3} - \frac{c}{c+3}\right) \cdot \frac{c-3}{c+1}\), we'll follow these steps: 1. **Find a common denominator** for the two fractions in the first part of the expression. 2. **Simplify the expression**. 3. **Multiply by \(\frac{c-3}{c+1}\)**. ### Step 1: Find a common denominator The common denominator for \(\frac{c+3}{c-3}\) and \(\frac{c}{c+3}\) is \((c-3)(c+3)\). Rewriting both fractions with a common denominator: \[ \frac{c+3}{c-3} = \frac{(c+3)(c+3)}{(c-3)(c+3)} \Rightarrow \frac{(c+3)^2}{(c-3)(c+3)} \] \[ \frac{c}{c+3} = \frac{c(c-3)}{(c+3)(c-3)} \Rightarrow \frac{c(c-3)}{(c-3)(c+3)} \] Now, we can perform the subtraction: \[ \frac{(c+3)^2 - c(c-3)}{(c-3)(c+3)} \] ### Step 2: Simplify Now we compute \((c+3)^2 - c(c-3)\): \[ (c+3)^2 = c^2 + 6c + 9 \] \[ c(c-3) = c^2 - 3c \] Combine these expressions: \[ (c^2 + 6c + 9) - (c^2 - 3c) = c^2 + 6c + 9 - c^2 + 3c = 9c + 9 \] Thus, we have: \[ \frac{9(c+1)}{(c-3)(c+3)} \] ### Step 3: Multiply by \(\frac{c-3}{c+1}\) Now we multiply this result by \(\frac{c-3}{c+1}\): \[ \left(\frac{9(c+1)}{(c-3)(c+3)}\right) \cdot \left(\frac{c-3}{c+1}\right) \] Notice that \((c+1)\) in the numerator and denominator cancels out, and also \((c-3)\) in the numerator and denominator cancels out: This gives us: \[ \frac{9}{c+3} \] ### Final Answer Therefore, the simplified form of the original expression is: \[ \frac{9}{c+3} \]