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Answer :
To solve the problem of identifying which expressions are sums of perfect cubes, it's helpful to know what a perfect cube looks like. A perfect cube is an expression that can be written as [tex]\( a^3 + b^3 \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are expressions themselves (which can include numbers, variables, or both).
Let's examine each expression to determine if it's a sum of perfect cubes:
1. [tex]\( 8x^6 + 27 \)[/tex]:
- We need to write it as [tex]\( a^3 + b^3 \)[/tex].
- [tex]\( a = (2x^2) \)[/tex] because [tex]\( (2x^2)^3 = 8x^6 \)[/tex].
- [tex]\( b = 3 \)[/tex] because [tex]\( 3^3 = 27 \)[/tex].
- This expression can be rewritten as [tex]\( (2x^2)^3 + 3^3 \)[/tex], which is a sum of cubes.
2. [tex]\( x^9 + 1 \)[/tex]:
- Write [tex]\( a^3 + b^3 \)[/tex].
- [tex]\( a = (x^3) \)[/tex] because [tex]\( (x^3)^3 = x^9 \)[/tex].
- [tex]\( b = 1 \)[/tex] because [tex]\( 1^3 = 1 \)[/tex].
- This expression can be rewritten as [tex]\( (x^3)^3 + 1^3 \)[/tex], which is a sum of cubes.
3. [tex]\( 81x^3 + 16x^6 \)[/tex]:
- We attempt to express each term as a cube.
- [tex]\( 81x^3 \)[/tex] doesn't fit the form [tex]\( (something)^3 \)[/tex].
- [tex]\( 16x^6 \)[/tex] would be [tex]\( (2x^2)^3 \)[/tex] but still can't be combined with [tex]\( 81x^3 \)[/tex] into a sum of two cubes.
- This cannot be expressed as [tex]\( a^3 + b^3 \)[/tex].
4. [tex]\( x^6 + x^3 \)[/tex]:
- Neither term can be easily written in the form [tex]\( (something)^3 \)[/tex] and combine in a way that creates [tex]\( a^3 + b^3 \)[/tex].
- This expression is not a sum of cubes.
5. [tex]\( 27x^9 + x^{12} \)[/tex]:
- Write [tex]\( a^3 + b^3 \)[/tex].
- [tex]\( a = (3x^3) \)[/tex] because [tex]\( (3x^3)^3 = 27x^9 \)[/tex].
- [tex]\( b = (x^4) \)[/tex] because [tex]\( (x^4)^3 = x^{12} \)[/tex].
- This expression can be rewritten as [tex]\( (3x^3)^3 + (x^4)^3 \)[/tex], which is a sum of cubes.
6. [tex]\( 9x^3 + 27x^9 \)[/tex]:
- Neither term can be easily written in the form [tex]\( (something)^3 \)[/tex] in a manner that creates [tex]\( a^3 + b^3 \)[/tex].
- This expression is not a sum of 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]\( 27x^9 + x^{12} \)[/tex]
Let's examine each expression to determine if it's a sum of perfect cubes:
1. [tex]\( 8x^6 + 27 \)[/tex]:
- We need to write it as [tex]\( a^3 + b^3 \)[/tex].
- [tex]\( a = (2x^2) \)[/tex] because [tex]\( (2x^2)^3 = 8x^6 \)[/tex].
- [tex]\( b = 3 \)[/tex] because [tex]\( 3^3 = 27 \)[/tex].
- This expression can be rewritten as [tex]\( (2x^2)^3 + 3^3 \)[/tex], which is a sum of cubes.
2. [tex]\( x^9 + 1 \)[/tex]:
- Write [tex]\( a^3 + b^3 \)[/tex].
- [tex]\( a = (x^3) \)[/tex] because [tex]\( (x^3)^3 = x^9 \)[/tex].
- [tex]\( b = 1 \)[/tex] because [tex]\( 1^3 = 1 \)[/tex].
- This expression can be rewritten as [tex]\( (x^3)^3 + 1^3 \)[/tex], which is a sum of cubes.
3. [tex]\( 81x^3 + 16x^6 \)[/tex]:
- We attempt to express each term as a cube.
- [tex]\( 81x^3 \)[/tex] doesn't fit the form [tex]\( (something)^3 \)[/tex].
- [tex]\( 16x^6 \)[/tex] would be [tex]\( (2x^2)^3 \)[/tex] but still can't be combined with [tex]\( 81x^3 \)[/tex] into a sum of two cubes.
- This cannot be expressed as [tex]\( a^3 + b^3 \)[/tex].
4. [tex]\( x^6 + x^3 \)[/tex]:
- Neither term can be easily written in the form [tex]\( (something)^3 \)[/tex] and combine in a way that creates [tex]\( a^3 + b^3 \)[/tex].
- This expression is not a sum of cubes.
5. [tex]\( 27x^9 + x^{12} \)[/tex]:
- Write [tex]\( a^3 + b^3 \)[/tex].
- [tex]\( a = (3x^3) \)[/tex] because [tex]\( (3x^3)^3 = 27x^9 \)[/tex].
- [tex]\( b = (x^4) \)[/tex] because [tex]\( (x^4)^3 = x^{12} \)[/tex].
- This expression can be rewritten as [tex]\( (3x^3)^3 + (x^4)^3 \)[/tex], which is a sum of cubes.
6. [tex]\( 9x^3 + 27x^9 \)[/tex]:
- Neither term can be easily written in the form [tex]\( (something)^3 \)[/tex] in a manner that creates [tex]\( a^3 + b^3 \)[/tex].
- This expression is not a sum of 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]\( 27x^9 + x^{12} \)[/tex]
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