(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:
- Find a common denominator for the two fractions in the first part of the expression.
- Simplify the expression.
- 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:
\[ \frac{9}{c+3} \]
