Answer :
Sure! Let's go through each expression and determine which are sums of perfect cubes.
1. Expression: [tex]\(8x^6 + 27\)[/tex]
- [tex]\(8x^6\)[/tex] can be rewritten as [tex]\((2x^2)^3\)[/tex] because [tex]\((2x^2)\)[/tex] cubed is [tex]\(8x^6\)[/tex].
- [tex]\(27\)[/tex] is a cube of [tex]\(3\)[/tex] because [tex]\(3^3 = 27\)[/tex].
- Therefore, [tex]\(8x^6 + 27\)[/tex] can be expressed as a sum of perfect cubes.
2. Expression: [tex]\(x^9 + 1\)[/tex]
- [tex]\(x^9\)[/tex] can be rewritten as [tex]\((x^3)^3\)[/tex] because [tex]\((x^3)\)[/tex] cubed is [tex]\(x^9\)[/tex].
- [tex]\(1\)[/tex] is a cube of [tex]\(1\)[/tex] because [tex]\(1^3 = 1\)[/tex].
- Therefore, [tex]\(x^9 + 1\)[/tex] is a sum of perfect cubes.
3. Expression: [tex]\(81x^3 + 16x^6\)[/tex]
- [tex]\(81x^3\)[/tex] can be rewritten as [tex]\((3x)^3\)[/tex] because [tex]\((3x)\)[/tex] cubed is [tex]\(81x^3\)[/tex].
- [tex]\(16x^6\)[/tex] can be rewritten as [tex]\((2x^2)^3\)[/tex] because [tex]\((2x^2)\)[/tex] cubed is [tex]\(16x^6\)[/tex].
- Therefore, [tex]\(81x^3 + 16x^6\)[/tex] is a sum of perfect cubes.
4. Expression: [tex]\(x^6 + x^3\)[/tex]
- These terms cannot be directly rewritten as a sum of two perfect cubes.
- Therefore, [tex]\(x^6 + x^3\)[/tex] is not a sum of perfect cubes.
5. Expression: [tex]\(27x^9 + x^{12}\)[/tex]
- [tex]\(27x^9\)[/tex] can be rewritten as [tex]\((3x^3)^3\)[/tex] because [tex]\((3x^3)\)[/tex] cubed is [tex]\(27x^9\)[/tex].
- [tex]\(x^{12}\)[/tex] can be rewritten as [tex]\((x^4)^3\)[/tex] because [tex]\((x^4)\)[/tex] cubed is [tex]\(x^{12}\)[/tex].
- Therefore, [tex]\(27x^9 + x^{12}\)[/tex] is a sum of perfect cubes.
6. Expression: [tex]\(9x^3 + 27x^9\)[/tex]
- These terms cannot be directly rewritten as a sum of two perfect cubes.
- Therefore, [tex]\(9x^3 + 27x^9\)[/tex] is not a sum of perfect cubes.
After analyzing each expression, the sums of perfect cubes are:
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(81x^3 + 16x^6\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
1. Expression: [tex]\(8x^6 + 27\)[/tex]
- [tex]\(8x^6\)[/tex] can be rewritten as [tex]\((2x^2)^3\)[/tex] because [tex]\((2x^2)\)[/tex] cubed is [tex]\(8x^6\)[/tex].
- [tex]\(27\)[/tex] is a cube of [tex]\(3\)[/tex] because [tex]\(3^3 = 27\)[/tex].
- Therefore, [tex]\(8x^6 + 27\)[/tex] can be expressed as a sum of perfect cubes.
2. Expression: [tex]\(x^9 + 1\)[/tex]
- [tex]\(x^9\)[/tex] can be rewritten as [tex]\((x^3)^3\)[/tex] because [tex]\((x^3)\)[/tex] cubed is [tex]\(x^9\)[/tex].
- [tex]\(1\)[/tex] is a cube of [tex]\(1\)[/tex] because [tex]\(1^3 = 1\)[/tex].
- Therefore, [tex]\(x^9 + 1\)[/tex] is a sum of perfect cubes.
3. Expression: [tex]\(81x^3 + 16x^6\)[/tex]
- [tex]\(81x^3\)[/tex] can be rewritten as [tex]\((3x)^3\)[/tex] because [tex]\((3x)\)[/tex] cubed is [tex]\(81x^3\)[/tex].
- [tex]\(16x^6\)[/tex] can be rewritten as [tex]\((2x^2)^3\)[/tex] because [tex]\((2x^2)\)[/tex] cubed is [tex]\(16x^6\)[/tex].
- Therefore, [tex]\(81x^3 + 16x^6\)[/tex] is a sum of perfect cubes.
4. Expression: [tex]\(x^6 + x^3\)[/tex]
- These terms cannot be directly rewritten as a sum of two perfect cubes.
- Therefore, [tex]\(x^6 + x^3\)[/tex] is not a sum of perfect cubes.
5. Expression: [tex]\(27x^9 + x^{12}\)[/tex]
- [tex]\(27x^9\)[/tex] can be rewritten as [tex]\((3x^3)^3\)[/tex] because [tex]\((3x^3)\)[/tex] cubed is [tex]\(27x^9\)[/tex].
- [tex]\(x^{12}\)[/tex] can be rewritten as [tex]\((x^4)^3\)[/tex] because [tex]\((x^4)\)[/tex] cubed is [tex]\(x^{12}\)[/tex].
- Therefore, [tex]\(27x^9 + x^{12}\)[/tex] is a sum of perfect cubes.
6. Expression: [tex]\(9x^3 + 27x^9\)[/tex]
- These terms cannot be directly rewritten as a sum of two perfect cubes.
- Therefore, [tex]\(9x^3 + 27x^9\)[/tex] is not a sum of perfect cubes.
After analyzing each expression, the sums of perfect cubes are:
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(81x^3 + 16x^6\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
