Answer :
To determine which expressions are sums of perfect cubes, we first need to identify if each term in the expression can be represented as a perfect cube. Let's go through each expression one by one:
1. [tex]\(8x^6 + 27\)[/tex]:
- [tex]\(8x^6 = (2x^2)^3\)[/tex] is a perfect cube.
- [tex]\(27 = 3^3\)[/tex] is a perfect cube.
- Therefore, [tex]\(8x^6 + 27\)[/tex] is a sum of perfect cubes.
2. [tex]\(x^9 + 1\)[/tex]:
- [tex]\(x^9 = (x^3)^3\)[/tex] is a perfect cube.
- [tex]\(1 = 1^3\)[/tex] is a perfect cube.
- Therefore, [tex]\(x^9 + 1\)[/tex] is a sum of perfect cubes.
3. [tex]\(81x^3 + 16x^6\)[/tex]:
- [tex]\(81x^3\)[/tex] can be rewritten as [tex]\((3x)^3\)[/tex] because [tex]\(81 = (3^3) = 27\)[/tex], not a perfect cube.
- [tex]\(16x^6 = (4x^2)^3\)[/tex] is a perfect cube, since [tex]\(16 = 2^4\)[/tex].
- The expression is not a sum of perfect cubes because [tex]\(81x^3\)[/tex] is incorrect.
4. [tex]\(x^6 + x^3\)[/tex]:
- [tex]\(x^6 = (x^2)^3\)[/tex] is a perfect cube.
- [tex]\(x^3 = (x)^3\)[/tex] is a perfect cube.
- Therefore, [tex]\(x^6 + x^3\)[/tex] is a sum of perfect cubes.
5. [tex]\(27x^9 + x^{12}\)[/tex]:
- [tex]\(27x^9 = (3x^3)^3\)[/tex] is a perfect cube.
- [tex]\(x^{12} = (x^4)^3\)[/tex] is a perfect cube.
- Therefore, [tex]\(27x^9 + x^{12}\)[/tex] is a sum of perfect cubes.
6. [tex]\(9x^3 + 27x^9\)[/tex]:
- [tex]\(9x^3\)[/tex] can be rewritten as [tex]\((3x)^3\)[/tex] because [tex]\(9 = 3^2\)[/tex], not a perfect cube.
- [tex]\(27x^9 = (3x^3)^3\)[/tex] is a perfect cube.
- The expression is not a sum of perfect cubes because [tex]\(9x^3\)[/tex] is not a perfect cube.
Summary of sums of perfect cubes:
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(x^6 + x^3\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
These are the expressions that are sums of perfect cubes.
1. [tex]\(8x^6 + 27\)[/tex]:
- [tex]\(8x^6 = (2x^2)^3\)[/tex] is a perfect cube.
- [tex]\(27 = 3^3\)[/tex] is a perfect cube.
- Therefore, [tex]\(8x^6 + 27\)[/tex] is a sum of perfect cubes.
2. [tex]\(x^9 + 1\)[/tex]:
- [tex]\(x^9 = (x^3)^3\)[/tex] is a perfect cube.
- [tex]\(1 = 1^3\)[/tex] is a perfect cube.
- Therefore, [tex]\(x^9 + 1\)[/tex] is a sum of perfect cubes.
3. [tex]\(81x^3 + 16x^6\)[/tex]:
- [tex]\(81x^3\)[/tex] can be rewritten as [tex]\((3x)^3\)[/tex] because [tex]\(81 = (3^3) = 27\)[/tex], not a perfect cube.
- [tex]\(16x^6 = (4x^2)^3\)[/tex] is a perfect cube, since [tex]\(16 = 2^4\)[/tex].
- The expression is not a sum of perfect cubes because [tex]\(81x^3\)[/tex] is incorrect.
4. [tex]\(x^6 + x^3\)[/tex]:
- [tex]\(x^6 = (x^2)^3\)[/tex] is a perfect cube.
- [tex]\(x^3 = (x)^3\)[/tex] is a perfect cube.
- Therefore, [tex]\(x^6 + x^3\)[/tex] is a sum of perfect cubes.
5. [tex]\(27x^9 + x^{12}\)[/tex]:
- [tex]\(27x^9 = (3x^3)^3\)[/tex] is a perfect cube.
- [tex]\(x^{12} = (x^4)^3\)[/tex] is a perfect cube.
- Therefore, [tex]\(27x^9 + x^{12}\)[/tex] is a sum of perfect cubes.
6. [tex]\(9x^3 + 27x^9\)[/tex]:
- [tex]\(9x^3\)[/tex] can be rewritten as [tex]\((3x)^3\)[/tex] because [tex]\(9 = 3^2\)[/tex], not a perfect cube.
- [tex]\(27x^9 = (3x^3)^3\)[/tex] is a perfect cube.
- The expression is not a sum of perfect cubes because [tex]\(9x^3\)[/tex] is not a perfect cube.
Summary of sums of perfect cubes:
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(x^6 + x^3\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
These are the expressions that are sums of perfect cubes.
