%5E3-%2B-(4x-2y)%5E3%3D16(ax%5E3%2Bbxy%5E2).jpeg "(4x+2y)^3 + (4x-2y)^3=16(ax^3+bxy^2)")
Expanding and Simplifying the Equation: (4x+2y)^3 + (4x-2y)^3 = 16(ax^3+bxy^2)
This problem involves expanding and simplifying a given algebraic expression to determine the values of constants 'a' and 'b'. Let's break down the steps:
1. Expanding the Cubes
We can use the following identity to expand the cubes: (a + b)³ = a³ + 3a²b + 3ab² + b³
Applying this to our equation:
(4x + 2y)³ = (4x)³ + 3(4x)²(2y) + 3(4x)(2y)² + (2y)³ = 64x³ + 96x²y + 48xy² + 8y³
(4x - 2y)³ = (4x)³ + 3(4x)²(-2y) + 3(4x)(-2y)² + (-2y)³ = 64x³ - 96x²y + 48xy² - 8y³
2. Adding the Expanded Terms
Now, let's add the two expanded expressions:
(64x³ + 96x²y + 48xy² + 8y³) + (64x³ - 96x²y + 48xy² - 8y³) = 128x³ + 96xy²
3. Simplifying and Comparing
The simplified expression on the left side is 128x³ + 96xy². Let's compare this to the right side of the original equation, 16(ax³ + bxy²):
128x³ + 96xy² = 16(ax³ + bxy²)
To make both sides equal, we need:
- 16a = 128
- 16b = 96
4. Solving for a and b
Dividing both sides of the equations by 16, we get:
- a = 8
- b = 6
Therefore, the values of a = 8 and b = 6 satisfy the given equation.