Answer :
To determine if an expression is a sum of two perfect cubes, we check each term to see if it can be written in the form
[tex]$$
(a x^m)^3 = a^3 \, x^{3m},
$$[/tex]
which means that the numerical coefficient is a perfect cube and the exponent on [tex]\(x\)[/tex] is a multiple of 3.
Let's analyze each expression step by step.
1. Expression 1: [tex]\(8x^6+27\)[/tex]
- [tex]\(8x^6\)[/tex]: The coefficient [tex]\(8\)[/tex] is a perfect cube since [tex]\(2^3=8\)[/tex], and the exponent [tex]\(6\)[/tex] is a multiple of 3 ([tex]\(6=3\cdot2\)[/tex]). Hence,
[tex]$$
8x^6 = (2x^2)^3.
$$[/tex]
- [tex]\(27\)[/tex]: The number [tex]\(27\)[/tex] is a perfect cube since [tex]\(3^3=27\)[/tex]. So,
[tex]$$
27 = 3^3.
$$[/tex]
- Both terms are perfect cubes. Thus, [tex]\(8x^6+27\)[/tex] is a sum of perfect cubes.
2. Expression 2: [tex]\(x^9+1\)[/tex]
- [tex]\(x^9\)[/tex]: The exponent [tex]\(9\)[/tex] is a multiple of 3 ([tex]\(9=3\cdot3\)[/tex]), so we can write
[tex]$$
x^9 = (x^3)^3.
$$[/tex]
- [tex]\(1\)[/tex]: The number [tex]\(1\)[/tex] is a perfect cube since [tex]\(1^3=1\)[/tex].
- Therefore, [tex]\(x^9+1\)[/tex] is a sum of perfect cubes.
3. Expression 3: [tex]\(81x^3+16x^6\)[/tex]
- [tex]\(81x^3\)[/tex]: The exponent [tex]\(3\)[/tex] is acceptable, but consider the coefficient [tex]\(81\)[/tex]. Although [tex]\(81\)[/tex] is a power of 3, it is not a perfect cube because no integer [tex]\(b\)[/tex] satisfies [tex]\(b^3=81\)[/tex] (since [tex]\(3^3=27\)[/tex] and [tex]\(4^3=64\)[/tex] while [tex]\(5^3=125\)[/tex]).
- [tex]\(16x^6\)[/tex]: The coefficient [tex]\(16\)[/tex] is not a perfect cube either (as [tex]\(2^3=8\)[/tex] and [tex]\(3^3=27\)[/tex]).
- Since one or both terms are not perfect cubes, [tex]\(81x^3+16x^6\)[/tex] is not a sum of perfect cubes.
4. Expression 4: [tex]\(x^6+x^3\)[/tex]
- [tex]\(x^6\)[/tex]: Since [tex]\(6\)[/tex] is a multiple of 3 ([tex]\(6=3\cdot2\)[/tex]), we have:
[tex]$$
x^6 = (x^2)^3.
$$[/tex]
- [tex]\(x^3\)[/tex]: Clearly,
[tex]$$
x^3 = (x)^3.
$$[/tex]
- Both terms are perfect cubes, so [tex]\(x^6+x^3\)[/tex] is the sum of two cubes.
5. Expression 5: [tex]\(27x^9+x^{12}\)[/tex]
- [tex]\(27x^9\)[/tex]: The coefficient [tex]\(27\)[/tex] is a perfect cube since [tex]\(3^3=27\)[/tex], and the exponent [tex]\(9\)[/tex] is a multiple of 3 ([tex]\(9=3\cdot3\)[/tex]). Hence,
[tex]$$
27x^9 = (3x^3)^3.
$$[/tex]
- [tex]\(x^{12}\)[/tex]: The exponent [tex]\(12\)[/tex] is a multiple of 3 ([tex]\(12=3\cdot4\)[/tex]) so we can write:
[tex]$$
x^{12} = (x^4)^3.
$$[/tex]
- Both terms are perfect cubes. Thus, [tex]\(27x^9+x^{12}\)[/tex] is a sum of perfect cubes.
6. Expression 6: [tex]\(9x^3+27x^9\)[/tex]
- [tex]\(9x^3\)[/tex]: While the exponent [tex]\(3\)[/tex] is fine, the coefficient [tex]\(9\)[/tex] is not a perfect cube (since [tex]\(2^3=8\)[/tex] and [tex]\(3^3=27\)[/tex]).
- [tex]\(27x^9\)[/tex]: This term is a perfect cube as explained before.
- Since [tex]\(9x^3\)[/tex] is not a perfect cube, [tex]\(9x^3+27x^9\)[/tex] is not a sum of perfect cubes.
Based on the analysis, the expressions that are sums of perfect cubes are those numbered:
[tex]$$
\text{1, 2, 4, and 5.}
$$[/tex]
[tex]$$
(a x^m)^3 = a^3 \, x^{3m},
$$[/tex]
which means that the numerical coefficient is a perfect cube and the exponent on [tex]\(x\)[/tex] is a multiple of 3.
Let's analyze each expression step by step.
1. Expression 1: [tex]\(8x^6+27\)[/tex]
- [tex]\(8x^6\)[/tex]: The coefficient [tex]\(8\)[/tex] is a perfect cube since [tex]\(2^3=8\)[/tex], and the exponent [tex]\(6\)[/tex] is a multiple of 3 ([tex]\(6=3\cdot2\)[/tex]). Hence,
[tex]$$
8x^6 = (2x^2)^3.
$$[/tex]
- [tex]\(27\)[/tex]: The number [tex]\(27\)[/tex] is a perfect cube since [tex]\(3^3=27\)[/tex]. So,
[tex]$$
27 = 3^3.
$$[/tex]
- Both terms are perfect cubes. Thus, [tex]\(8x^6+27\)[/tex] is a sum of perfect cubes.
2. Expression 2: [tex]\(x^9+1\)[/tex]
- [tex]\(x^9\)[/tex]: The exponent [tex]\(9\)[/tex] is a multiple of 3 ([tex]\(9=3\cdot3\)[/tex]), so we can write
[tex]$$
x^9 = (x^3)^3.
$$[/tex]
- [tex]\(1\)[/tex]: The number [tex]\(1\)[/tex] is a perfect cube since [tex]\(1^3=1\)[/tex].
- Therefore, [tex]\(x^9+1\)[/tex] is a sum of perfect cubes.
3. Expression 3: [tex]\(81x^3+16x^6\)[/tex]
- [tex]\(81x^3\)[/tex]: The exponent [tex]\(3\)[/tex] is acceptable, but consider the coefficient [tex]\(81\)[/tex]. Although [tex]\(81\)[/tex] is a power of 3, it is not a perfect cube because no integer [tex]\(b\)[/tex] satisfies [tex]\(b^3=81\)[/tex] (since [tex]\(3^3=27\)[/tex] and [tex]\(4^3=64\)[/tex] while [tex]\(5^3=125\)[/tex]).
- [tex]\(16x^6\)[/tex]: The coefficient [tex]\(16\)[/tex] is not a perfect cube either (as [tex]\(2^3=8\)[/tex] and [tex]\(3^3=27\)[/tex]).
- Since one or both terms are not perfect cubes, [tex]\(81x^3+16x^6\)[/tex] is not a sum of perfect cubes.
4. Expression 4: [tex]\(x^6+x^3\)[/tex]
- [tex]\(x^6\)[/tex]: Since [tex]\(6\)[/tex] is a multiple of 3 ([tex]\(6=3\cdot2\)[/tex]), we have:
[tex]$$
x^6 = (x^2)^3.
$$[/tex]
- [tex]\(x^3\)[/tex]: Clearly,
[tex]$$
x^3 = (x)^3.
$$[/tex]
- Both terms are perfect cubes, so [tex]\(x^6+x^3\)[/tex] is the sum of two cubes.
5. Expression 5: [tex]\(27x^9+x^{12}\)[/tex]
- [tex]\(27x^9\)[/tex]: The coefficient [tex]\(27\)[/tex] is a perfect cube since [tex]\(3^3=27\)[/tex], and the exponent [tex]\(9\)[/tex] is a multiple of 3 ([tex]\(9=3\cdot3\)[/tex]). Hence,
[tex]$$
27x^9 = (3x^3)^3.
$$[/tex]
- [tex]\(x^{12}\)[/tex]: The exponent [tex]\(12\)[/tex] is a multiple of 3 ([tex]\(12=3\cdot4\)[/tex]) so we can write:
[tex]$$
x^{12} = (x^4)^3.
$$[/tex]
- Both terms are perfect cubes. Thus, [tex]\(27x^9+x^{12}\)[/tex] is a sum of perfect cubes.
6. Expression 6: [tex]\(9x^3+27x^9\)[/tex]
- [tex]\(9x^3\)[/tex]: While the exponent [tex]\(3\)[/tex] is fine, the coefficient [tex]\(9\)[/tex] is not a perfect cube (since [tex]\(2^3=8\)[/tex] and [tex]\(3^3=27\)[/tex]).
- [tex]\(27x^9\)[/tex]: This term is a perfect cube as explained before.
- Since [tex]\(9x^3\)[/tex] is not a perfect cube, [tex]\(9x^3+27x^9\)[/tex] is not a sum of perfect cubes.
Based on the analysis, the expressions that are sums of perfect cubes are those numbered:
[tex]$$
\text{1, 2, 4, and 5.}
$$[/tex]
