Answer :
To determine which expressions are sums of perfect cubes, we need to identify whether each expression can be rewritten as a sum of two cubed terms. A sum of cubes generally takes the form:
[tex]\[ a^3 + b^3 \][/tex]
Let's evaluate each expression individually:
1. Expression: [tex]\(8x^6 + 27\)[/tex]
- We look for terms that can be expressed as cubes.
- [tex]\(8x^6 = (2x^2)^3\)[/tex] and [tex]\(27 = 3^3\)[/tex].
- This expression can be written as a sum of cubes: [tex]\((2x^2)^3 + 3^3\)[/tex].
2. Expression: [tex]\(x^9 + 1\)[/tex]
- We need terms that are perfect cubes.
- [tex]\(x^9 = (x^3)^3\)[/tex] and [tex]\(1 = 1^3\)[/tex].
- This expression can be written as a sum of cubes: [tex]\((x^3)^3 + 1^3\)[/tex].
3. Expression: [tex]\(81x^3 + 16x^6\)[/tex]
- Check for perfect cubes.
- [tex]\(81x^3 = (3x)^3\)[/tex] and [tex]\(16x^6 = (2x^2)^3\)[/tex].
- This expression is a sum of cubes: [tex]\((3x)^3 + (2x^2)^3\)[/tex].
4. Expression: [tex]\(x^6 + x^3\)[/tex]
- Try to express terms as cubes.
- [tex]\(x^6 = (x^2)^3\)[/tex] is a perfect cube.
- [tex]\(x^3\)[/tex] is already a cube: [tex]\((x)^3\)[/tex].
- So, it can also be written as: [tex]\((x^2)^3 + (x)^3\)[/tex].
5. Expression: [tex]\(27x^9 + x^{12}\)[/tex]
- Determine the cube form.
- [tex]\(27x^9 = (3x^3)^3\)[/tex] and [tex]\(x^{12} = (x^4)^3\)[/tex].
- This expression is a sum of cubes: [tex]\((3x^3)^3 + (x^4)^3\)[/tex].
6. Expression: [tex]\(9x^3 + 27x^9\)[/tex]
- Verify cube conditions.
- [tex]\(9x^3\)[/tex] cannot directly be expressed as a cube, as [tex]\(9\)[/tex] is not a perfect cube.
- Similarly, [tex]\(27x^9 = (3x^3)^3\)[/tex], which is a perfect cube.
- Since we can’t express both terms as cubes, this is not a sum of cubes.
After a careful evaluation, none of these expressions can adequately be expressed as a sum of perfect cubes. So, it turns out that none of the expressions match the form [tex]\(a^3 + b^3\)[/tex] after appropriate transformations.
Thus, none of these are sums of perfect cubes.
[tex]\[ a^3 + b^3 \][/tex]
Let's evaluate each expression individually:
1. Expression: [tex]\(8x^6 + 27\)[/tex]
- We look for terms that can be expressed as cubes.
- [tex]\(8x^6 = (2x^2)^3\)[/tex] and [tex]\(27 = 3^3\)[/tex].
- This expression can be written as a sum of cubes: [tex]\((2x^2)^3 + 3^3\)[/tex].
2. Expression: [tex]\(x^9 + 1\)[/tex]
- We need terms that are perfect cubes.
- [tex]\(x^9 = (x^3)^3\)[/tex] and [tex]\(1 = 1^3\)[/tex].
- This expression can be written as a sum of cubes: [tex]\((x^3)^3 + 1^3\)[/tex].
3. Expression: [tex]\(81x^3 + 16x^6\)[/tex]
- Check for perfect cubes.
- [tex]\(81x^3 = (3x)^3\)[/tex] and [tex]\(16x^6 = (2x^2)^3\)[/tex].
- This expression is a sum of cubes: [tex]\((3x)^3 + (2x^2)^3\)[/tex].
4. Expression: [tex]\(x^6 + x^3\)[/tex]
- Try to express terms as cubes.
- [tex]\(x^6 = (x^2)^3\)[/tex] is a perfect cube.
- [tex]\(x^3\)[/tex] is already a cube: [tex]\((x)^3\)[/tex].
- So, it can also be written as: [tex]\((x^2)^3 + (x)^3\)[/tex].
5. Expression: [tex]\(27x^9 + x^{12}\)[/tex]
- Determine the cube form.
- [tex]\(27x^9 = (3x^3)^3\)[/tex] and [tex]\(x^{12} = (x^4)^3\)[/tex].
- This expression is a sum of cubes: [tex]\((3x^3)^3 + (x^4)^3\)[/tex].
6. Expression: [tex]\(9x^3 + 27x^9\)[/tex]
- Verify cube conditions.
- [tex]\(9x^3\)[/tex] cannot directly be expressed as a cube, as [tex]\(9\)[/tex] is not a perfect cube.
- Similarly, [tex]\(27x^9 = (3x^3)^3\)[/tex], which is a perfect cube.
- Since we can’t express both terms as cubes, this is not a sum of cubes.
After a careful evaluation, none of these expressions can adequately be expressed as a sum of perfect cubes. So, it turns out that none of the expressions match the form [tex]\(a^3 + b^3\)[/tex] after appropriate transformations.
Thus, none of these are sums of perfect cubes.
