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

- [tex]8x^6 + 27[/tex]
- [tex]x^9 + 1[/tex]
- [tex]81x^3 + 16x^6[/tex]
- [tex]x^6 + x^3[/tex]
- [tex]27x^9 + x^{12}[/tex]
- [tex]9x^3 + 27x^9[/tex]

Answer :

To determine which expressions are sums of perfect cubes, we need to express each term in the expressions as cubes of simpler expressions. A perfect cube is a term that can be expressed in the form [tex]\( a^3 \)[/tex]. A sum of perfect cubes is an expression like [tex]\( a^3 + b^3 \)[/tex].

Let's check each expression one by one:

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

- [tex]\( 8x^6 \)[/tex] can be written as [tex]\( (2x^2)^3 \)[/tex]
- [tex]\( 27 \)[/tex] can be written as [tex]\( 3^3 \)[/tex]

Thus, [tex]\( 8x^6 + 27 = (2x^2)^3 + 3^3 \)[/tex] is a sum of perfect cubes.

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

- [tex]\( x^9 \)[/tex] can be written as [tex]\( (x^3)^3 \)[/tex]
- [tex]\( 1 \)[/tex] can be written as [tex]\( 1^3 \)[/tex]

Hence, [tex]\( x^9 + 1 = (x^3)^3 + 1^3 \)[/tex] is a sum of perfect cubes.

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

- [tex]\( 81x^3 \)[/tex] can be written as [tex]\( (3x)^3 \)[/tex]
- [tex]\( 16x^6 \)[/tex] can be written as [tex]\( (2x^2)^3 \)[/tex]

Thus, [tex]\( 81x^3 + 16x^6 = (3x)^3 + (2x^2)^3 \)[/tex] is a sum of perfect cubes.

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

- [tex]\( x^6 = (x^2)^3 \)[/tex]
- [tex]\( x^3 \neq (\text{perfect cube})^3 \)[/tex] so [tex]\( x^3 \)[/tex] cannot be written as a cube.

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 written as [tex]\( (3x^3)^3 \)[/tex]
- [tex]\( x^{12} \)[/tex] can be written as [tex]\( (x^4)^3 \)[/tex]

Thus, [tex]\( 27x^9 + x^{12} = (3x^3)^3 + (x^4)^3 \)[/tex] is a sum of perfect cubes.

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

- [tex]\( 9x^3 \neq (\text{perfect cube})^3 \)[/tex] so it cannot be written as a cube.
- [tex]\( 27x^9 = (3x^3)^3 \)[/tex]

Because [tex]\( 9x^3 \)[/tex] is not a cube, [tex]\( 9x^3 + 27x^9 \)[/tex] is not a sum of perfect cubes.

In summary, the expressions that are 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]