Answer :
To determine which expressions are sums of perfect cubes, we need to identify if each expression can be written in the form [tex]\( a^3 + b^3 \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are algebraic expressions themselves.
Let's break down the provided expressions one by one:
1. Expression 1: [tex]\( 8x^6 + 27 \)[/tex]
- This expression can be written as [tex]\( (2x^2)^3 + 3^3 \)[/tex].
- It matches the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = 2x^2 \)[/tex] and [tex]\( b = 3 \)[/tex].
- Therefore, this is a sum of perfect cubes.
2. Expression 2: [tex]\( x^9 + 1 \)[/tex]
- We need to find [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that [tex]\( a^3 = x^9 \)[/tex] and [tex]\( b^3 = 1 \)[/tex].
- This would require [tex]\( a = x^3 \)[/tex] and [tex]\( b = 1 \)[/tex], which does match the form.
- Therefore, this is a sum of perfect cubes: [tex]\( (x^3)^3 + 1^3 \)[/tex].
3. Expression 3: [tex]\( 81x^3 + 16x^6 \)[/tex]
- This cannot be written in the form of a sum of perfect cubes [tex]\( a^3 + b^3 \)[/tex].
- Neither term alone is a perfect cube, nor can they be paired to create [tex]\( a^3 + b^3 \)[/tex].
- Thus, it is not a sum of perfect cubes.
4. Expression 4: [tex]\( x^6 + x^3 \)[/tex]
- For [tex]\( x^6 \)[/tex] to be a perfect cube, it would need to be [tex]\( (x^2)^3 \)[/tex] and [tex]\( x^3 \)[/tex] would need to be a perfect cube which is [tex]\( (x)^3 \)[/tex].
- Here the whole expression doesn't take the form [tex]\( a^3 + b^3 \properly.
- Therefore, this is not a sum of perfect cubes.
5. Expression 5: \( 27x^9 + x^{12} \)[/tex]
- This can be rewritten as [tex]\( (3x^3)^3 + (x^4)^3 \)[/tex].
- It matches the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = 3x^3 \)[/tex] and [tex]\( b = x^4 \)[/tex].
- Hence, it is a sum of perfect cubes.
6. Expression 6: [tex]\( 9x^3 + 27x^9 \)[/tex]
- This expression can be rewritten as [tex]\( (x^3)^3 + (3x^3)^3 \)[/tex].
- It matches the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = x^3 \)[/tex] and [tex]\( b = 3x^3 \)[/tex].
- Therefore, this is a sum of perfect cubes.
So, the expressions that are sums of perfect cubes are [tex]\( 8x^6 + 27 \)[/tex], [tex]\( x^9 + 1 \)[/tex], [tex]\( 27x^9 + x^{12} \)[/tex], and [tex]\( 9x^3 + 27x^9 \)[/tex].
Let's break down the provided expressions one by one:
1. Expression 1: [tex]\( 8x^6 + 27 \)[/tex]
- This expression can be written as [tex]\( (2x^2)^3 + 3^3 \)[/tex].
- It matches the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = 2x^2 \)[/tex] and [tex]\( b = 3 \)[/tex].
- Therefore, this is a sum of perfect cubes.
2. Expression 2: [tex]\( x^9 + 1 \)[/tex]
- We need to find [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that [tex]\( a^3 = x^9 \)[/tex] and [tex]\( b^3 = 1 \)[/tex].
- This would require [tex]\( a = x^3 \)[/tex] and [tex]\( b = 1 \)[/tex], which does match the form.
- Therefore, this is a sum of perfect cubes: [tex]\( (x^3)^3 + 1^3 \)[/tex].
3. Expression 3: [tex]\( 81x^3 + 16x^6 \)[/tex]
- This cannot be written in the form of a sum of perfect cubes [tex]\( a^3 + b^3 \)[/tex].
- Neither term alone is a perfect cube, nor can they be paired to create [tex]\( a^3 + b^3 \)[/tex].
- Thus, it is not a sum of perfect cubes.
4. Expression 4: [tex]\( x^6 + x^3 \)[/tex]
- For [tex]\( x^6 \)[/tex] to be a perfect cube, it would need to be [tex]\( (x^2)^3 \)[/tex] and [tex]\( x^3 \)[/tex] would need to be a perfect cube which is [tex]\( (x)^3 \)[/tex].
- Here the whole expression doesn't take the form [tex]\( a^3 + b^3 \properly.
- Therefore, this is not a sum of perfect cubes.
5. Expression 5: \( 27x^9 + x^{12} \)[/tex]
- This can be rewritten as [tex]\( (3x^3)^3 + (x^4)^3 \)[/tex].
- It matches the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = 3x^3 \)[/tex] and [tex]\( b = x^4 \)[/tex].
- Hence, it is a sum of perfect cubes.
6. Expression 6: [tex]\( 9x^3 + 27x^9 \)[/tex]
- This expression can be rewritten as [tex]\( (x^3)^3 + (3x^3)^3 \)[/tex].
- It matches the form [tex]\( a^3 + b^3 \)[/tex] with [tex]\( a = x^3 \)[/tex] and [tex]\( b = 3x^3 \)[/tex].
- Therefore, this is a sum of perfect cubes.
So, the expressions that are sums of perfect cubes are [tex]\( 8x^6 + 27 \)[/tex], [tex]\( x^9 + 1 \)[/tex], [tex]\( 27x^9 + x^{12} \)[/tex], and [tex]\( 9x^3 + 27x^9 \)[/tex].
