Multiply or divide, and then simplify the following expressions:
1. \(\frac{(x+1)^2}{x-5} \cdot \frac{x-3}{x+1}\)
2. \(\frac{3x+3}{2} \div \frac{4}{9x+9}\)
3. \(\frac{4a}{3a-3} \cdot \frac{3(a-1)}{16a}\)
4. \(\frac{x+2}{x-1} \div \frac{5x+10}{x-2}\)
Answer (2)
x − 5 ( x + 1 ) 2 ⋅ x + 1 x − 3 = x − 5 x + 1 ⋅ 1 x − 3 = x − 5 ( x + 1 ) ( x − 3 ) = x − 5 x 2 − 3 x + x − 3 = x − 5 x 2 − 2 x − 3 D : x = 5 ∧ x = − 1
2 3 x + 3 : 9 x + 9 4 = 2 3 x + 3 ⋅ 4 9 x + 9 = 8 27 x 2 + 27 x + 27 x + 27 = 8 27 x 2 + 54 x + 27 D : x = 1
3 a − 3 4 a ⋅ 16 a 3 ( a − 1 ) = 3 a − 3 1 ⋅ 4 a 3 a − 3 = 4 a 1 D : a = 1 ∧ a = 0
x − 1 x + 2 : x − 2 5 x + 10 = x − 1 x + 2 ⋅ 5 x + 10 x − 2 = x − 1 x + 2 ⋅ 5 ( x + 2 ) x − 2 = x − 1 1 ⋅ 5 x − 2 = 5 x − 5 x − 2 D : x = 1 ∧ x = 2 ∧ x = − 2
The expressions were simplified step by step by canceling common factors and changing division to multiplication. Each expression has been simplified to its final form with noted domain restrictions. This ensures the solutions are accurate and relevant based on algebraic principles.
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