Answer :
To determine which expressions are sums of perfect cubes, we need to check if each expression can be written in the form [tex]\( a^3 + b^3 \)[/tex]. Let's break it down step by step for each expression:
1. Expression: [tex]\(8x^6 + 27\)[/tex]
Decompose this expression:
- Recognize: [tex]\( 8x^6 = (2x^2)^3 \)[/tex]
- Recognize: [tex]\( 27 = 3^3 \)[/tex]
- Therefore, the expression is in the form of [tex]\( a^3 + b^3 \)[/tex] where [tex]\( a = 2x^2 \)[/tex] and [tex]\( b = 3 \)[/tex].
2. Expression: [tex]\(x^9 + 1\)[/tex]
Decompose this expression:
- Recognize: [tex]\( x^9 = (x^3)^3 \)[/tex]
- Recognize: [tex]\( 1 = 1^3 \)[/tex]
- Therefore, the expression is in the form of [tex]\( a^3 + b^3 \)[/tex] where [tex]\( a = x^3 \)[/tex] and [tex]\( b = 1 \)[/tex].
3. Expression: [tex]\(81x^3 + 16x^6\)[/tex]
Decompose this expression:
- Recognize: [tex]\( 81x^3 = (3x)^3 \)[/tex]
- Recognize: [tex]\( 16x^6 = (2x^2)^3 \)[/tex]
- Therefore, the expression is in the form of [tex]\( a^3 + b^3 \)[/tex] where [tex]\( a = 3x \)[/tex] and [tex]\( b = 2x^2 \)[/tex].
4. Expression: [tex]\(x^6 + x^3\)[/tex]
Decompose this expression:
- Recognize: [tex]\( x^6 = (x^2)^3 \)[/tex]
- Recognize: [tex]\( x^3 = (x)^3 \)[/tex]
- Therefore, the expression is in the form of [tex]\( a^3 + b^3 \)[/tex] where [tex]\( a = x^2 \)[/tex] and [tex]\( b = x \)[/tex].
5. Expression: [tex]\(27x^9 + x^{12}\)[/tex]
Decompose this expression:
- Recognize: [tex]\( 27x^9 = (3x^3)^3 \)[/tex]
- Recognize: [tex]\( x^{12} = (x^4)^3 \)[/tex]
- Therefore, the expression is in the form of [tex]\( a^3 + b^3 \)[/tex] where [tex]\( a = 3x^3 \)[/tex] and [tex]\( b = x^4 \)[/tex].
6. Expression: [tex]\(9x^3 + 27x^9\)[/tex]
Attempt to decompose this expression:
- Recognize: It involves terms [tex]\( x^3 \)[/tex] and [tex]\( x^9 \)[/tex], but they do not directly match a structure suitable for a sum of cubes like the others.
- There is no straightforward way to express [tex]\( 9x^3 \)[/tex] itself as a perfect cube.
After examining all the expressions, the ones that can be decomposed into 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]
Thus, the correct sums of perfect cubes are those represented by indices [1, 2, 3, 4, 5].
1. Expression: [tex]\(8x^6 + 27\)[/tex]
Decompose this expression:
- Recognize: [tex]\( 8x^6 = (2x^2)^3 \)[/tex]
- Recognize: [tex]\( 27 = 3^3 \)[/tex]
- Therefore, the expression is in the form of [tex]\( a^3 + b^3 \)[/tex] where [tex]\( a = 2x^2 \)[/tex] and [tex]\( b = 3 \)[/tex].
2. Expression: [tex]\(x^9 + 1\)[/tex]
Decompose this expression:
- Recognize: [tex]\( x^9 = (x^3)^3 \)[/tex]
- Recognize: [tex]\( 1 = 1^3 \)[/tex]
- Therefore, the expression is in the form of [tex]\( a^3 + b^3 \)[/tex] where [tex]\( a = x^3 \)[/tex] and [tex]\( b = 1 \)[/tex].
3. Expression: [tex]\(81x^3 + 16x^6\)[/tex]
Decompose this expression:
- Recognize: [tex]\( 81x^3 = (3x)^3 \)[/tex]
- Recognize: [tex]\( 16x^6 = (2x^2)^3 \)[/tex]
- Therefore, the expression is in the form of [tex]\( a^3 + b^3 \)[/tex] where [tex]\( a = 3x \)[/tex] and [tex]\( b = 2x^2 \)[/tex].
4. Expression: [tex]\(x^6 + x^3\)[/tex]
Decompose this expression:
- Recognize: [tex]\( x^6 = (x^2)^3 \)[/tex]
- Recognize: [tex]\( x^3 = (x)^3 \)[/tex]
- Therefore, the expression is in the form of [tex]\( a^3 + b^3 \)[/tex] where [tex]\( a = x^2 \)[/tex] and [tex]\( b = x \)[/tex].
5. Expression: [tex]\(27x^9 + x^{12}\)[/tex]
Decompose this expression:
- Recognize: [tex]\( 27x^9 = (3x^3)^3 \)[/tex]
- Recognize: [tex]\( x^{12} = (x^4)^3 \)[/tex]
- Therefore, the expression is in the form of [tex]\( a^3 + b^3 \)[/tex] where [tex]\( a = 3x^3 \)[/tex] and [tex]\( b = x^4 \)[/tex].
6. Expression: [tex]\(9x^3 + 27x^9\)[/tex]
Attempt to decompose this expression:
- Recognize: It involves terms [tex]\( x^3 \)[/tex] and [tex]\( x^9 \)[/tex], but they do not directly match a structure suitable for a sum of cubes like the others.
- There is no straightforward way to express [tex]\( 9x^3 \)[/tex] itself as a perfect cube.
After examining all the expressions, the ones that can be decomposed into 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]
Thus, the correct sums of perfect cubes are those represented by indices [1, 2, 3, 4, 5].
