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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 :

Sure, let's look at each expression to find out which ones are sums of perfect cubes.

1. Expression: [tex]\(8x^6 + 27\)[/tex]

We can express [tex]\(8x^6\)[/tex] as [tex]\((2x^2)^3\)[/tex] and [tex]\(27\)[/tex] as [tex]\(3^3\)[/tex]. Therefore, this expression can be written as the sum of two perfect cubes: [tex]\((2x^2)^3 + 3^3\)[/tex].

2. Expression: [tex]\(x^9 + 1\)[/tex]

Here, [tex]\(x^9\)[/tex] can be rewritten as [tex]\((x^3)^3\)[/tex] and [tex]\(1\)[/tex] is [tex]\(1^3\)[/tex]. Thus, this expression is [tex]\((x^3)^3 + 1^3\)[/tex], making it the sum of two perfect cubes.

3. Expression: [tex]\(81x^3 + 16x^6\)[/tex]

In this case, [tex]\(81x^3\)[/tex] can be expressed as [tex]\((3x)^3\)[/tex] and [tex]\(16x^6\)[/tex] as [tex]\((2x^2)^3\)[/tex]. So, the expression is [tex]\((3x)^3 + (2x^2)^3\)[/tex], representing a sum of two perfect cubes.

4. Expression: [tex]\(x^6 + x^3\)[/tex]

This expression does not fit the form of a sum of two perfect cubes. Therefore, it cannot be written in such a form.

5. Expression: [tex]\(27x^9 + x^{12}\)[/tex]

We can express [tex]\(27x^9\)[/tex] as [tex]\((3x^3)^3\)[/tex] and [tex]\(x^{12}\)[/tex] as [tex]\((x^4)^3\)[/tex]. Thus, this expression can be written as the sum of two perfect cubes: [tex]\((3x^3)^3 + (x^4)^3\)[/tex].

6. Expression: [tex]\(9x^3 + 27x^9\)[/tex]

This expression cannot be simplified into the form of a sum of two perfect cubes. Hence, it is not a sum of perfect cubes.

Based on these evaluations, the sums of perfect cubes from the options are:

- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]

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