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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 need to look at each term in the expression and verify if they can be written as a perfect cube.

### 1. [tex]\(8x^6 + 27\)[/tex]
- [tex]\(8x^6 = (2x^2)^3\)[/tex] (perfect cube)
- [tex]\(27 = 3^3\)[/tex] (perfect cube)
- This is a sum of perfect cubes.

### 2. [tex]\(x^9 + 1\)[/tex]
- [tex]\(x^9 = (x^3)^3\)[/tex] (perfect cube)
- [tex]\(1 = 1^3\)[/tex] (perfect cube)
- This is a sum of perfect cubes.

### 3. [tex]\(81x^3 + 16x^6\)[/tex]
- [tex]\(81x^3 = (3^4 x)^3\)[/tex] however, [tex]\(3^4 = 81\)[/tex] is not a cube,
- [tex]\(16x^6 = (4x^2)^3\)[/tex], however, neither 4 nor [tex]\(4x^2\)[/tex] form a cube.
- This is not a sum of perfect cubes.

### 4. [tex]\(x^6 + x^3\)[/tex]
- [tex]\(x^6 = (x^2)^3\)[/tex] (perfect cube)
- [tex]\(x^3 = x^3\)[/tex] (perfect cube)
- This is a sum of perfect cubes.

### 5. [tex]\(27x^9 + x^{12}\)[/tex]
- [tex]\(27x^9 = (3x^3)^3\)[/tex] (perfect cube)
- [tex]\(x^{12} = (x^4)^3\)[/tex] (perfect cube)
- This is a sum of perfect cubes.

### 6. [tex]\(9x^3 + 27x^9\)[/tex]
- [tex]\(9x^3 = (3x)^3\)[/tex], [tex]\(9\)[/tex] is not a cube
- [tex]\(27x^9 = (3x^3)^3\)[/tex] (perfect cube)
- This is not a sum of perfect cubes.

### Final Answer
The sums of perfect cubes are:
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(x^6 + x^3\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]

Therefore, you should check the following:
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(x^6 + x^3\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]

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