Answer :
To answer the question of which expression is a sum of cubes, we need to understand some basic properties of cubes and sums of cubes.
First, let's recall the form of a sum of cubes. A sum of cubes expression has the form [tex]\(a^3 + b^3\)[/tex], where both [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are terms that are each raised to the power of 3.
Given the expressions:
1. [tex]\(-64 x^6 y^{12} + 125 x^{16} y^3\)[/tex]
2. [tex]\(-32 x^6 y^{12} + 125 x^{16} y^3\)[/tex]
3. [tex]\(32 x^6 y^{12} + 125 x^9 y^3\)[/tex]
4. [tex]\(64 x^6 y^{12} + 125 x^9 y^3\)[/tex]
### Analyzing Each Expression:
#### Expression 1: [tex]\(-64 x^6 y^{12} + 125 x^{16} y^3\)[/tex]
- For [tex]\(-64 x^6 y^{12}\)[/tex], rewrite as [tex]\(- (4 x^2 y^4)^3\)[/tex]
- For [tex]\(125 x^{16} y^3\)[/tex], rewrite as [tex]\((5 x^{16/3} y) y\neq0 (but it does not comply with a perfect cube format like \(a^3 + b^3\)[/tex].
#### Expression 2: [tex]\(-32 x^6 y^{12} + 125 x^{16} y^3\)[/tex]
- Similar issues arise since [tex]\(-32 x^6 y^{12}\)[/tex] can not be expressed perfectly in cube:
and thus [tex]\((5 x^{16/3} y) y\neq0 (but it does not comply with a perfect cube format like \(a^3 + b^3\)[/tex].
#### Expression 3: [tex]\(32 x^6 y^{12} + 125 x^9 y^3\)[/tex]
- Again, [tex]\(32 x^6 y^{12}\)[/tex] can be expressed as [tex]\((2 x^2 y^4)^3\)[/tex]
- [tex]\(125 x^9 y^3\)[/tex] cannot be rewritten in a way complying with the cube format.
#### Expression 4: [tex]\(64 x^6 y^{12} + 125 x^9 y^3\)[/tex]
- [tex]\(64 x^6 y^{12}\)[/tex] can be rewritten as [tex]\((4 x^2 y^4)^3\)[/tex]
- [tex]\(125 x^9 y^3\)[/tex] also mismatches cube compliance
### Conclusion
After examining the given expressions, none of them can be expressed as a sum of cubes ([tex]\(a^3 + b^3\)[/tex]).
Thus, none of the given expressions is a sum of cubes.
First, let's recall the form of a sum of cubes. A sum of cubes expression has the form [tex]\(a^3 + b^3\)[/tex], where both [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are terms that are each raised to the power of 3.
Given the expressions:
1. [tex]\(-64 x^6 y^{12} + 125 x^{16} y^3\)[/tex]
2. [tex]\(-32 x^6 y^{12} + 125 x^{16} y^3\)[/tex]
3. [tex]\(32 x^6 y^{12} + 125 x^9 y^3\)[/tex]
4. [tex]\(64 x^6 y^{12} + 125 x^9 y^3\)[/tex]
### Analyzing Each Expression:
#### Expression 1: [tex]\(-64 x^6 y^{12} + 125 x^{16} y^3\)[/tex]
- For [tex]\(-64 x^6 y^{12}\)[/tex], rewrite as [tex]\(- (4 x^2 y^4)^3\)[/tex]
- For [tex]\(125 x^{16} y^3\)[/tex], rewrite as [tex]\((5 x^{16/3} y) y\neq0 (but it does not comply with a perfect cube format like \(a^3 + b^3\)[/tex].
#### Expression 2: [tex]\(-32 x^6 y^{12} + 125 x^{16} y^3\)[/tex]
- Similar issues arise since [tex]\(-32 x^6 y^{12}\)[/tex] can not be expressed perfectly in cube:
and thus [tex]\((5 x^{16/3} y) y\neq0 (but it does not comply with a perfect cube format like \(a^3 + b^3\)[/tex].
#### Expression 3: [tex]\(32 x^6 y^{12} + 125 x^9 y^3\)[/tex]
- Again, [tex]\(32 x^6 y^{12}\)[/tex] can be expressed as [tex]\((2 x^2 y^4)^3\)[/tex]
- [tex]\(125 x^9 y^3\)[/tex] cannot be rewritten in a way complying with the cube format.
#### Expression 4: [tex]\(64 x^6 y^{12} + 125 x^9 y^3\)[/tex]
- [tex]\(64 x^6 y^{12}\)[/tex] can be rewritten as [tex]\((4 x^2 y^4)^3\)[/tex]
- [tex]\(125 x^9 y^3\)[/tex] also mismatches cube compliance
### Conclusion
After examining the given expressions, none of them can be expressed as a sum of cubes ([tex]\(a^3 + b^3\)[/tex]).
Thus, none of the given expressions is a sum of cubes.
