Answer :
Sure, let's go through each expression step-by-step to determine which ones are sums of perfect cubes.
1. Expression: [tex]\(8x^6 + 27\)[/tex]
- We break down the terms: [tex]\(8x^6\)[/tex] is [tex]\((2x^2)^3\)[/tex] and [tex]\(27\)[/tex] is [tex]\(3^3\)[/tex].
- This fits the formula for the sum of cubes: [tex]\((a^3 + b^3)\)[/tex].
- Therefore, [tex]\(8x^6 + 27\)[/tex] is a sum of perfect cubes.
2. Expression: [tex]\(x^9 + 1\)[/tex]
- Here, [tex]\(x^9\)[/tex] can be written as [tex]\((x^3)^3\)[/tex] and [tex]\(1\)[/tex] as [tex]\(1^3\)[/tex].
- This fits the sum of cubes format [tex]\((a^3 + b^3)\)[/tex].
- So, [tex]\(x^9 + 1\)[/tex] is a sum of perfect cubes.
3. Expression: [tex]\(81x^3 + 16x^6\)[/tex]
- We can write [tex]\(81x^3\)[/tex] as [tex]\((3x)^3\)[/tex] and [tex]\(16x^6\)[/tex] as [tex]\((2x^2)^3\)[/tex].
- This expression fits the sum of cubes pattern [tex]\((a^3 + b^3)\)[/tex].
- Hence, [tex]\(81x^3 + 16x^6\)[/tex] is a sum of perfect cubes.
4. Expression: [tex]\(x^6 + x^3\)[/tex]
- Break it down: [tex]\(x^6\)[/tex] is [tex]\((x^2)^3\)[/tex], and [tex]\(x^3\)[/tex] is just a single cube term.
- However, we need something of the format [tex]\((a^3 + b^3)\)[/tex], and this doesn’t fit that exactly.
- So, this is not a sum of perfect cubes.
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].
- This matches the sum of cubes formula [tex]\((a^3 + b^3)\)[/tex].
- Therefore, [tex]\(27x^9 + x^{12}\)[/tex] is a sum of perfect cubes.
6. Expression: [tex]\(9x^3 + 27x^9\)[/tex]
- Factor: [tex]\(9x^3(1 + 3x^6)\)[/tex].
- After factoring, this expression does not fit the pattern needed for a sum of cubes.
- Thus, [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], and [tex]\(27x^9 + x^{12}\)[/tex]. Therefore, the correct indices are [1, 2, 3, 5].
1. Expression: [tex]\(8x^6 + 27\)[/tex]
- We break down the terms: [tex]\(8x^6\)[/tex] is [tex]\((2x^2)^3\)[/tex] and [tex]\(27\)[/tex] is [tex]\(3^3\)[/tex].
- This fits the formula for the sum of cubes: [tex]\((a^3 + b^3)\)[/tex].
- Therefore, [tex]\(8x^6 + 27\)[/tex] is a sum of perfect cubes.
2. Expression: [tex]\(x^9 + 1\)[/tex]
- Here, [tex]\(x^9\)[/tex] can be written as [tex]\((x^3)^3\)[/tex] and [tex]\(1\)[/tex] as [tex]\(1^3\)[/tex].
- This fits the sum of cubes format [tex]\((a^3 + b^3)\)[/tex].
- So, [tex]\(x^9 + 1\)[/tex] is a sum of perfect cubes.
3. Expression: [tex]\(81x^3 + 16x^6\)[/tex]
- We can write [tex]\(81x^3\)[/tex] as [tex]\((3x)^3\)[/tex] and [tex]\(16x^6\)[/tex] as [tex]\((2x^2)^3\)[/tex].
- This expression fits the sum of cubes pattern [tex]\((a^3 + b^3)\)[/tex].
- Hence, [tex]\(81x^3 + 16x^6\)[/tex] is a sum of perfect cubes.
4. Expression: [tex]\(x^6 + x^3\)[/tex]
- Break it down: [tex]\(x^6\)[/tex] is [tex]\((x^2)^3\)[/tex], and [tex]\(x^3\)[/tex] is just a single cube term.
- However, we need something of the format [tex]\((a^3 + b^3)\)[/tex], and this doesn’t fit that exactly.
- So, this is not a sum of perfect cubes.
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].
- This matches the sum of cubes formula [tex]\((a^3 + b^3)\)[/tex].
- Therefore, [tex]\(27x^9 + x^{12}\)[/tex] is a sum of perfect cubes.
6. Expression: [tex]\(9x^3 + 27x^9\)[/tex]
- Factor: [tex]\(9x^3(1 + 3x^6)\)[/tex].
- After factoring, this expression does not fit the pattern needed for a sum of cubes.
- Thus, [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], and [tex]\(27x^9 + x^{12}\)[/tex]. Therefore, the correct indices are [1, 2, 3, 5].
