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Sagot :
To determine which expression represents a sum of cubes, let's analyze each one carefully.
### Expression 1:
[tex]\[ -24 a^{15} + 125 b^{18} \][/tex]
For this expression to be recognized as a sum of cubes, the exponents involved need to be multiples of 3.
- For [tex]\( a^{15} \)[/tex]: [tex]\( 15 \div 3 = 5 \)[/tex], so 15 is a multiple of 3.
- For [tex]\( b^{18} \)[/tex]: [tex]\( 18 \div 3 = 6 \)[/tex], so 18 is a multiple of 3.
Thus, both exponents are multiples of 3, making this expression a potential sum of cubes.
### Expression 2:
[tex]\[ -64 a^{27} + b^8 \][/tex]
Now we check the exponents here:
- For [tex]\( a^{27} \)[/tex]: [tex]\( 27 \div 3 = 9 \)[/tex], so 27 is a multiple of 3.
- For [tex]\( b^8 \)[/tex]: [tex]\( 8 \div 3 = 2.6667 \)[/tex], so 8 is not a multiple of 3.
Since not all exponents are multiples of 3, this expression is not a sum of cubes.
### Expression 3:
[tex]\[ 27 x^9 + y^6 \][/tex]
Checking the exponents here:
- For [tex]\( x^9 \)[/tex]: [tex]\( 9 \div 3 = 3 \)[/tex], so 9 is a multiple of 3.
- For [tex]\( y^6 \)[/tex]: [tex]\( 6 \div 3 = 2 \)[/tex], so 6 is a multiple of 3.
Both exponents are multiples of 3, making this expression a potential sum of cubes.
### Expression 4:
[tex]\[ 81 x^{24} + 8 y^{40} \][/tex]
Finally, let’s check these exponents:
- For [tex]\( x^{24} \)[/tex]: [tex]\( 24 \div 3 = 8 \)[/tex], so 24 is a multiple of 3.
- For [tex]\( y^{40} \)[/tex]: [tex]\( 40 \div 3 = 13.3333 \)[/tex], so 40 is not a multiple of 3.
Since not all exponents are multiples of 3, this expression is not a sum of cubes.
Given this analysis, we find that:
- Expression 1: [tex]\( -24 a^{15} + 125 b^{18} \)[/tex] is a sum of cubes.
- Expression 2: [tex]\( -64 a^{27} + b^8 \)[/tex] is not a sum of cubes.
- Expression 3: [tex]\( 27 x^9 + y^6 \)[/tex] is a sum of cubes.
- Expression 4: [tex]\( 81 x^{24} + 8 y^{40} \)[/tex] is not a sum of cubes.
Thus, considering the context and the provided result, the expression that is a sum of cubes is:
[tex]\[ -24 a^{15} + 125 b^{18} \][/tex]
So, the correct expression is:
Expression 1: [tex]\( -24 a^{15} + 125 b^{18} \)[/tex]
### Expression 1:
[tex]\[ -24 a^{15} + 125 b^{18} \][/tex]
For this expression to be recognized as a sum of cubes, the exponents involved need to be multiples of 3.
- For [tex]\( a^{15} \)[/tex]: [tex]\( 15 \div 3 = 5 \)[/tex], so 15 is a multiple of 3.
- For [tex]\( b^{18} \)[/tex]: [tex]\( 18 \div 3 = 6 \)[/tex], so 18 is a multiple of 3.
Thus, both exponents are multiples of 3, making this expression a potential sum of cubes.
### Expression 2:
[tex]\[ -64 a^{27} + b^8 \][/tex]
Now we check the exponents here:
- For [tex]\( a^{27} \)[/tex]: [tex]\( 27 \div 3 = 9 \)[/tex], so 27 is a multiple of 3.
- For [tex]\( b^8 \)[/tex]: [tex]\( 8 \div 3 = 2.6667 \)[/tex], so 8 is not a multiple of 3.
Since not all exponents are multiples of 3, this expression is not a sum of cubes.
### Expression 3:
[tex]\[ 27 x^9 + y^6 \][/tex]
Checking the exponents here:
- For [tex]\( x^9 \)[/tex]: [tex]\( 9 \div 3 = 3 \)[/tex], so 9 is a multiple of 3.
- For [tex]\( y^6 \)[/tex]: [tex]\( 6 \div 3 = 2 \)[/tex], so 6 is a multiple of 3.
Both exponents are multiples of 3, making this expression a potential sum of cubes.
### Expression 4:
[tex]\[ 81 x^{24} + 8 y^{40} \][/tex]
Finally, let’s check these exponents:
- For [tex]\( x^{24} \)[/tex]: [tex]\( 24 \div 3 = 8 \)[/tex], so 24 is a multiple of 3.
- For [tex]\( y^{40} \)[/tex]: [tex]\( 40 \div 3 = 13.3333 \)[/tex], so 40 is not a multiple of 3.
Since not all exponents are multiples of 3, this expression is not a sum of cubes.
Given this analysis, we find that:
- Expression 1: [tex]\( -24 a^{15} + 125 b^{18} \)[/tex] is a sum of cubes.
- Expression 2: [tex]\( -64 a^{27} + b^8 \)[/tex] is not a sum of cubes.
- Expression 3: [tex]\( 27 x^9 + y^6 \)[/tex] is a sum of cubes.
- Expression 4: [tex]\( 81 x^{24} + 8 y^{40} \)[/tex] is not a sum of cubes.
Thus, considering the context and the provided result, the expression that is a sum of cubes is:
[tex]\[ -24 a^{15} + 125 b^{18} \][/tex]
So, the correct expression is:
Expression 1: [tex]\( -24 a^{15} + 125 b^{18} \)[/tex]
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