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Answer :
To determine which of the given expressions are sums of perfect cubes, we need to check if each expression can be written in the form [tex]\(a^3 + b^3\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are expressions.
Let's go through each expression:
1. [tex]\(8x^6 + 27\)[/tex]
- This can be factored into [tex]\((2x^2)^3 + 3^3\)[/tex].
- Therefore, [tex]\(8x^6 + 27\)[/tex] is a sum of perfect cubes.
2. [tex]\(x^9 + 1\)[/tex]
- This can be rewritten as [tex]\((x^3)^3 + 1^3\)[/tex].
- Therefore, [tex]\(x^9 + 1\)[/tex] is a sum of perfect cubes.
3. [tex]\(81x^3 + 16x^6\)[/tex]
- First term [tex]\(81x^3\)[/tex] is not a perfect cube. Second term [tex]\(16x^6\)[/tex] can be seen as [tex]\((2x^2)^3\)[/tex], but [tex]\(81x^3\)[/tex] is not a cube.
- Therefore, [tex]\(81x^3 + 16x^6\)[/tex] is not a sum of perfect cubes.
4. [tex]\(x^6 + x^3\)[/tex]
- Cannot be expressed in the form of [tex]\(a^3 + b^3\)[/tex].
- Therefore, [tex]\(x^6 + x^3\)[/tex] is not a sum of perfect cubes.
5. [tex]\(27x^9 + x^{12}\)[/tex]
- This can be rewritten as [tex]\((3x^3)^3 + (x^4)^3\)[/tex].
- Therefore, [tex]\(27x^9 + x^{12}\)[/tex] is a sum of perfect cubes.
6. [tex]\(9x^3 + 27x^9\)[/tex]
- Cannot be expressed in the form of [tex]\(a^3 + b^3\)[/tex].
- Therefore, [tex]\(9x^3 + 27x^9\)[/tex] is not a sum of perfect cubes.
Based on this analysis, the expressions that are sums of perfect cubes are:
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
Let's go through each expression:
1. [tex]\(8x^6 + 27\)[/tex]
- This can be factored into [tex]\((2x^2)^3 + 3^3\)[/tex].
- Therefore, [tex]\(8x^6 + 27\)[/tex] is a sum of perfect cubes.
2. [tex]\(x^9 + 1\)[/tex]
- This can be rewritten as [tex]\((x^3)^3 + 1^3\)[/tex].
- Therefore, [tex]\(x^9 + 1\)[/tex] is a sum of perfect cubes.
3. [tex]\(81x^3 + 16x^6\)[/tex]
- First term [tex]\(81x^3\)[/tex] is not a perfect cube. Second term [tex]\(16x^6\)[/tex] can be seen as [tex]\((2x^2)^3\)[/tex], but [tex]\(81x^3\)[/tex] is not a cube.
- Therefore, [tex]\(81x^3 + 16x^6\)[/tex] is not a sum of perfect cubes.
4. [tex]\(x^6 + x^3\)[/tex]
- Cannot be expressed in the form of [tex]\(a^3 + b^3\)[/tex].
- Therefore, [tex]\(x^6 + x^3\)[/tex] is not a sum of perfect cubes.
5. [tex]\(27x^9 + x^{12}\)[/tex]
- This can be rewritten as [tex]\((3x^3)^3 + (x^4)^3\)[/tex].
- Therefore, [tex]\(27x^9 + x^{12}\)[/tex] is a sum of perfect cubes.
6. [tex]\(9x^3 + 27x^9\)[/tex]
- Cannot be expressed in the form of [tex]\(a^3 + b^3\)[/tex].
- Therefore, [tex]\(9x^3 + 27x^9\)[/tex] is not a sum of perfect cubes.
Based on this analysis, the expressions that are sums of perfect cubes are:
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
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