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Which expression is a sum of cubes?

A. [tex]\(-24a^{15} + 125b^{18}\)[/tex]
B. [tex]\(-64a^{27} + b^8\)[/tex]
C. [tex]\(27x^9 + y^6\)[/tex]
D. [tex]\(81x^{24} + 8y^{40}\)[/tex]

Sagot :

GinnyAnswer
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]
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