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Answer :
To determine which expressions are sums of perfect cubes, let's analyze each expression. A perfect cube is a term that can be expressed as something raised to the power of 3, such as [tex]\(a^3\)[/tex].
1. [tex]\(x^6 + x^3\)[/tex]:
- [tex]\(x^6\)[/tex] can be written as [tex]\((x^2)^3\)[/tex], which is a perfect cube.
- [tex]\(x^3\)[/tex] is also a perfect cube.
- Conclusion: [tex]\(x^6 + x^3\)[/tex] is a sum of perfect cubes.
2. [tex]\(8x^6 + 27\)[/tex]:
- [tex]\(8x^6\)[/tex] can be written as [tex]\((2x^2)^3\)[/tex].
- [tex]\(27\)[/tex] can be written as [tex]\(3^3\)[/tex].
- Conclusion: [tex]\(8x^6 + 27\)[/tex] is a sum of perfect cubes.
3. [tex]\(9x^3 + 27x^9\)[/tex]:
- [tex]\(9x^3\)[/tex] cannot be written as a perfect cube because there isn't a simple cubic root for 9.
- Since the number itself is not a perfect cube, [tex]\(9x^3\)[/tex] is not a perfect cube.
- Therefore, this expression is not a sum of perfect cubes.
4. [tex]\(81x^3 + 16x^6\)[/tex]:
- [tex]\(81x^3\)[/tex] can be written as [tex]\((3x)^3\)[/tex].
- [tex]\(16x^6\)[/tex] can be written as [tex]\((2x^2)^3\)[/tex].
- Conclusion: [tex]\(81x^3 + 16x^6\)[/tex] is a sum of perfect cubes.
5. [tex]\(x^9 + 1\)[/tex]:
- [tex]\(x^9\)[/tex] can be written as [tex]\((x^3)^3\)[/tex], which is a perfect cube.
- [tex]\(1\)[/tex] is also a perfect cube, as [tex]\(1^3 = 1\)[/tex].
- Conclusion: [tex]\(x^9 + 1\)[/tex] is a sum of perfect cubes.
6. [tex]\(27x^9 + x^{12}\)[/tex]:
- [tex]\(27x^9\)[/tex] can be written as [tex]\((3x^3)^3\)[/tex].
- [tex]\(x^{12}\)[/tex] can be written as [tex]\((x^4)^3\)[/tex].
- Conclusion: [tex]\(27x^9 + x^{12}\)[/tex] is a sum of perfect cubes.
Based on this analysis, the expressions that are sums of perfect cubes are:
- [tex]\(x^6 + x^3\)[/tex]
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(81x^3 + 16x^6\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
These conclusions show that five of the provided expressions can be written as sums of perfect cubes.
1. [tex]\(x^6 + x^3\)[/tex]:
- [tex]\(x^6\)[/tex] can be written as [tex]\((x^2)^3\)[/tex], which is a perfect cube.
- [tex]\(x^3\)[/tex] is also a perfect cube.
- Conclusion: [tex]\(x^6 + x^3\)[/tex] is a sum of perfect cubes.
2. [tex]\(8x^6 + 27\)[/tex]:
- [tex]\(8x^6\)[/tex] can be written as [tex]\((2x^2)^3\)[/tex].
- [tex]\(27\)[/tex] can be written as [tex]\(3^3\)[/tex].
- Conclusion: [tex]\(8x^6 + 27\)[/tex] is a sum of perfect cubes.
3. [tex]\(9x^3 + 27x^9\)[/tex]:
- [tex]\(9x^3\)[/tex] cannot be written as a perfect cube because there isn't a simple cubic root for 9.
- Since the number itself is not a perfect cube, [tex]\(9x^3\)[/tex] is not a perfect cube.
- Therefore, this expression is not a sum of perfect cubes.
4. [tex]\(81x^3 + 16x^6\)[/tex]:
- [tex]\(81x^3\)[/tex] can be written as [tex]\((3x)^3\)[/tex].
- [tex]\(16x^6\)[/tex] can be written as [tex]\((2x^2)^3\)[/tex].
- Conclusion: [tex]\(81x^3 + 16x^6\)[/tex] is a sum of perfect cubes.
5. [tex]\(x^9 + 1\)[/tex]:
- [tex]\(x^9\)[/tex] can be written as [tex]\((x^3)^3\)[/tex], which is a perfect cube.
- [tex]\(1\)[/tex] is also a perfect cube, as [tex]\(1^3 = 1\)[/tex].
- Conclusion: [tex]\(x^9 + 1\)[/tex] is a sum of perfect cubes.
6. [tex]\(27x^9 + x^{12}\)[/tex]:
- [tex]\(27x^9\)[/tex] can be written as [tex]\((3x^3)^3\)[/tex].
- [tex]\(x^{12}\)[/tex] can be written as [tex]\((x^4)^3\)[/tex].
- Conclusion: [tex]\(27x^9 + x^{12}\)[/tex] is a sum of perfect cubes.
Based on this analysis, the expressions that are sums of perfect cubes are:
- [tex]\(x^6 + x^3\)[/tex]
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(81x^3 + 16x^6\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
These conclusions show that five of the provided expressions can be written as sums of perfect cubes.
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