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Which are sums of perfect cubes? Check all that apply.

A. [tex]8x^6 + 27[/tex]
B. [tex]x^9 + 1[/tex]
C. [tex]81x^3 + 16x^6[/tex]
D. [tex]x^6 + x^3[/tex]
E. [tex]27x^9 + x^{12}[/tex]
F. [tex]9x^3 + 27x^9[/tex]

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.

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