How can 125x^9 + 64y^12 be expressed as a sum of cubes?

To express 125x⁹ + 64y¹² as a sum of cubes, we can follow the formula for factoring a sum of cubes, which states:

• A³ + B³ = (A + B)(A² – AB + B²)

In this case, we need to identify the cubic terms:

1. Recognize that 125 is 5³ and 64 is 4³.
2. Thus, we can rewrite:
• 125x⁹ = (5x³)³
• 64y¹² = (4y⁴)³

From this, let:

• A = 5x³
• B = 4y⁴

We can plug these values into our sum of cubes formula:

125x⁹ + 64y¹² = (5x³ + 4y⁴)( (5x³)² – (5x³)(4y⁴) + (4y⁴)²)

Next, we calculate:

• (5x³)² = 25x⁶
• (5x³)(4y⁴) = 20x³y⁴
• (4y⁴)² = 16y⁸

Putting it all together, we have:

125x⁹ + 64y¹² = (5x³ + 4y⁴)(25x⁶ – 20x³y⁴ + 16y⁸)

This is the expression of 125x⁹ + 64y¹² as a sum of cubes.