To express the expression 125x9 + 64y12 as a sum of cubes, we can utilize the formula for a sum of cubes, which is given by:
- a3 + b3 = (a + b)(a2 – ab + b2)
In our case, we can identify the components:
- a = 5x3 (since (5x3)3 = 125x9)
- b = 4y4 (since (4y4)3 = 64y12)
Now we can rewrite the expression:
- 125x9 + 64y12 = (5x3)3 + (4y4)3
According to the sum of cubes formula, we have:
- (5x3 + 4y4)( (5x3)2 – (5x3)(4y4) + (4y4)2)
Next, we need to simplify the second part:
- (5x3)2 = 25x6
- (5x3)(4y4) = 20x3y4
- (4y4)2 = 16y8
Putting it all together, we can express the entire expression as:
- 125x9 + 64y12 = (5x3 + 4y4)(25x6 – 20x3y4 + 16y8)
This shows that 125x9 + 64y12 can indeed be expressed as a sum of cubes.