Question

A³-b³ . (a+b+a-b): a²+ab+b²=? ------- ------------ ---------------- 2a+2b (a-b+a+b) : a²-b² =?

Answer 1

$\frac{ {a}^{3} - {b}^{3} }{2a + 2b} \times \frac{(a + b + a - b)}{(a - b + a + b)} \div \frac{ {a}^{2} + ab + {b}^{2} }{ {a}^{2} - {b}^{2} } = \frac{ ({a}^{3} - {b}^{3}) \times (a - b + a + b) \times (a + b)(a - b) }{2(a + b) \times (a - b + a + b) \times ({a}^{2} + ab + {b}^{2}) } = \frac{(a - b)( {a}^{2} + ab + {b}^{2}) \times (a - b)}{2 \times ( {a}^{2} + ab + {b}^{2}) } = \frac{(a - b) \times (a - b)}{2} = \frac{ {(a - b})^{2} }{2} = \frac{ {a}^{2} - 2ab + {b}^{2} }{2}$

Answer 2

Let's simplify the expressions step-by-step:

1. Simplify (A³ - B³) / ((A + B + A - B) / (A² + AB + B²)):

Step 1: Factor the numerator (A³ - B³): A³ - B³ = (A - B)(A² + AB + B²)

Step 2: Simplify the denominator (A + B + A - B): A + B + A - B = 2A

Step 3: Simplify the expression: (A³ - B³) / ((A + B + A - B) / (A² + AB + B²)) = ((A - B)(A² + AB + B²)) / (2A / (A² + AB + B²))

Step 4: Invert the division and multiply: = ((A - B)(A² + AB + B²)) * ((A² + AB + B²) / 2A)

Step 5: Cancel common factors: = (A - B) * ((A² + AB + B²) / 2)

The simplified expression is: (A - B) * ((A² + AB + B²) / 2)

2. Simplify ((A - B + A + B) / (A² - B²)):

Step 1: Combine like terms in the numerator: (A - B + A + B) = 2A

Step 2: Factor the denominator (A² - B²): A² - B² = (A + B)(A - B)

Step 3: Simplify the expression: ((A - B + A + B) / (A² - B²)) = (2A / (A + B)(A - B))

The simplified expression is: 2A / (A + B)(A - B)