Answer :
To determine which expressions are sums of perfect cubes, we must first understand what a perfect cube is. A perfect cube is a number or expression that can be written as something raised to the power of 3, like [tex]\( (a)^3 \)[/tex].
Let's analyze each expression one by one:
1. [tex]\(8x^6 + 27\)[/tex]:
- [tex]\(8x^6\)[/tex] can be written as [tex]\((2x^2)^3\)[/tex], and
- [tex]\(27\)[/tex] can be written as [tex]\(3^3\)[/tex].
- So, this expression is a sum of two cubes, [tex]\((2x^2)^3 + 3^3\)[/tex].
2. [tex]\(x^9 + 1\)[/tex]:
- [tex]\(x^9\)[/tex] can be written as [tex]\((x^3)^3\)[/tex], and
- [tex]\(1\)[/tex] can be written as [tex]\(1^3\)[/tex].
- So, this expression is a sum of two cubes, [tex]\((x^3)^3 + 1^3\)[/tex].
3. [tex]\(81x^3 + 16x^6\)[/tex]:
- [tex]\(81x^3\)[/tex] can be written as [tex]\((3x)^3\)[/tex], and
- [tex]\(16x^6\)[/tex] can be written as [tex]\((2x^2)^3\)[/tex].
- So, this is a sum of two cubes, [tex]\((3x)^3 + (2x^2)^3\)[/tex].
4. [tex]\(x^6 + x^3\)[/tex]:
- [tex]\(x^6\)[/tex] can be written as [tex]\((x^2)^3\)[/tex].
- [tex]\(x^3\)[/tex] can be written as [tex]\((x)^3\)[/tex].
- This is a sum of two cubes, [tex]\((x^2)^3 + (x)^3\)[/tex].
5. [tex]\(27x^9 + x^{12}\)[/tex]:
- [tex]\(27x^9\)[/tex] can be written as [tex]\((3x^3)^3\)[/tex], and
- [tex]\(x^{12}\)[/tex] can be written as [tex]\((x^4)^3\)[/tex].
- So, this is a sum of two cubes, [tex]\((3x^3)^3 + (x^4)^3\)[/tex].
6. [tex]\(9x^3 + 27x^9\)[/tex]:
- [tex]\(9x^3\)[/tex] is not a perfect cube itself because it cannot be written as something cubed. [tex]\((3x)^3 = 27x^3\)[/tex], and
- [tex]\(27x^9\)[/tex] can be written as [tex]\((3x^3)^3\)[/tex].
- Therefore, we can't express this entire expression as a sum of perfect cubes.
Based on this analysis, 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]\(x^6 + x^3\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
These expressions can all be represented as sums of two perfect cubes.
Let's analyze each expression one by one:
1. [tex]\(8x^6 + 27\)[/tex]:
- [tex]\(8x^6\)[/tex] can be written as [tex]\((2x^2)^3\)[/tex], and
- [tex]\(27\)[/tex] can be written as [tex]\(3^3\)[/tex].
- So, this expression is a sum of two cubes, [tex]\((2x^2)^3 + 3^3\)[/tex].
2. [tex]\(x^9 + 1\)[/tex]:
- [tex]\(x^9\)[/tex] can be written as [tex]\((x^3)^3\)[/tex], and
- [tex]\(1\)[/tex] can be written as [tex]\(1^3\)[/tex].
- So, this expression is a sum of two cubes, [tex]\((x^3)^3 + 1^3\)[/tex].
3. [tex]\(81x^3 + 16x^6\)[/tex]:
- [tex]\(81x^3\)[/tex] can be written as [tex]\((3x)^3\)[/tex], and
- [tex]\(16x^6\)[/tex] can be written as [tex]\((2x^2)^3\)[/tex].
- So, this is a sum of two cubes, [tex]\((3x)^3 + (2x^2)^3\)[/tex].
4. [tex]\(x^6 + x^3\)[/tex]:
- [tex]\(x^6\)[/tex] can be written as [tex]\((x^2)^3\)[/tex].
- [tex]\(x^3\)[/tex] can be written as [tex]\((x)^3\)[/tex].
- This is a sum of two cubes, [tex]\((x^2)^3 + (x)^3\)[/tex].
5. [tex]\(27x^9 + x^{12}\)[/tex]:
- [tex]\(27x^9\)[/tex] can be written as [tex]\((3x^3)^3\)[/tex], and
- [tex]\(x^{12}\)[/tex] can be written as [tex]\((x^4)^3\)[/tex].
- So, this is a sum of two cubes, [tex]\((3x^3)^3 + (x^4)^3\)[/tex].
6. [tex]\(9x^3 + 27x^9\)[/tex]:
- [tex]\(9x^3\)[/tex] is not a perfect cube itself because it cannot be written as something cubed. [tex]\((3x)^3 = 27x^3\)[/tex], and
- [tex]\(27x^9\)[/tex] can be written as [tex]\((3x^3)^3\)[/tex].
- Therefore, we can't express this entire expression as a sum of perfect cubes.
Based on this analysis, 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]\(x^6 + x^3\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
These expressions can all be represented as sums of two perfect cubes.
