Answer :
We want to determine which expressions can be written as a sum of two perfect cubes. This means each term in the expression must be a perfect cube; that is, it can be expressed in the form $(\text{something})^3$.
Let’s analyze each option:
1. For
$$8x^6 + 27,$$
notice that
$$8x^6 = (2x^2)^3 \quad \text{and} \quad 27 = 3^3.$$
Both terms are perfect cubes, so this expression is a sum of perfect cubes.
2. For
$$x^9 + 1,$$
observe that
$$x^9 = (x^3)^3 \quad \text{and} \quad 1 = 1^3.$$
Both terms are perfect cubes, hence this is a sum of perfect cubes.
3. For
$$81x^3 + 16x^6,$$
notice that the coefficients $81$ and $16$ are not perfect cubes (since, for instance, $4^3 = 64$ and $5^3 = 125$, and neither $81$ nor $16$ equals any of these). Therefore, this expression cannot be written as a sum of perfect cubes.
4. For
$$x^6 + x^3,$$
we can write:
$$x^6 = (x^2)^3 \quad \text{and} \quad x^3 = (x)^3.$$
Both terms are perfect cubes, so this is a sum of perfect cubes.
5. For
$$27x^9 + x^{12},$$
we can express the terms as:
$$27x^9 = (3x^3)^3 \quad \text{and} \quad x^{12} = (x^4)^3.$$
Both terms are perfect cubes, meaning this is a sum of perfect cubes.
6. For
$$9x^3 + 27x^9,$$
observe that while
$$27x^9 = (3x^3)^3,$$
the term $9x^3$ does not correspond to a cube of a simple expression. Therefore, this is not a sum of perfect cubes.
Thus, the expressions that are sums of perfect cubes are those given in options 1, 2, 4, and 5.
Let’s analyze each option:
1. For
$$8x^6 + 27,$$
notice that
$$8x^6 = (2x^2)^3 \quad \text{and} \quad 27 = 3^3.$$
Both terms are perfect cubes, so this expression is a sum of perfect cubes.
2. For
$$x^9 + 1,$$
observe that
$$x^9 = (x^3)^3 \quad \text{and} \quad 1 = 1^3.$$
Both terms are perfect cubes, hence this is a sum of perfect cubes.
3. For
$$81x^3 + 16x^6,$$
notice that the coefficients $81$ and $16$ are not perfect cubes (since, for instance, $4^3 = 64$ and $5^3 = 125$, and neither $81$ nor $16$ equals any of these). Therefore, this expression cannot be written as a sum of perfect cubes.
4. For
$$x^6 + x^3,$$
we can write:
$$x^6 = (x^2)^3 \quad \text{and} \quad x^3 = (x)^3.$$
Both terms are perfect cubes, so this is a sum of perfect cubes.
5. For
$$27x^9 + x^{12},$$
we can express the terms as:
$$27x^9 = (3x^3)^3 \quad \text{and} \quad x^{12} = (x^4)^3.$$
Both terms are perfect cubes, meaning this is a sum of perfect cubes.
6. For
$$9x^3 + 27x^9,$$
observe that while
$$27x^9 = (3x^3)^3,$$
the term $9x^3$ does not correspond to a cube of a simple expression. Therefore, this is not a sum of perfect cubes.
Thus, the expressions that are sums of perfect cubes are those given in options 1, 2, 4, and 5.
