Answer :
To determine which of the given expressions are sums of perfect cubes, let's first understand what a perfect cube is. A number or term is a perfect cube if it can be expressed as something raised to the power of 3. In order to identify whether an expression is a sum of perfect cubes, it should fit the form [tex]\(a^3 + b^3\)[/tex].
Now, let's examine each of the expressions provided:
1. [tex]\(8x^6 + 27\)[/tex]
- The terms are [tex]\(8x^6\)[/tex] and [tex]\(27\)[/tex].
- [tex]\(8x^6\)[/tex] can be written as [tex]\((2x^2)^3\)[/tex] because [tex]\(8 = 2^3\)[/tex] and [tex]\((x^2)^3 = x^6\)[/tex].
- [tex]\(27\)[/tex] is [tex]\(3^3\)[/tex].
- Therefore, this is a sum of perfect cubes: [tex]\((2x^2)^3 + 3^3\)[/tex].
2. [tex]\(x^9 + 1\)[/tex]
- The terms are [tex]\(x^9\)[/tex] and [tex]\(1\)[/tex].
- [tex]\(x^9\)[/tex] can be written as [tex]\((x^3)^3\)[/tex].
- [tex]\(1\)[/tex] is [tex]\(1^3\)[/tex].
- Therefore, this is a sum of perfect cubes: [tex]\((x^3)^3 + 1^3\)[/tex].
3. [tex]\(81x^3 + 16x^6\)[/tex]
- The terms are [tex]\(81x^3\)[/tex] and [tex]\(16x^6\)[/tex].
- [tex]\(81x^3\)[/tex] can be written as [tex]\((3x)^3\)[/tex] because [tex]\(81 = 3^4\)[/tex] and [tex]\(x^3\)[/tex].
- [tex]\(16x^6\)[/tex] can be written as [tex]\((2x^2)^3\)[/tex] because [tex]\(16 = 2^4\)[/tex].
- Therefore, this is a sum of perfect cubes: [tex]\((3x)^3 + (2x^2)^3\)[/tex].
4. [tex]\(x^6 + x^3\)[/tex]
- The terms are [tex]\(x^6\)[/tex] and [tex]\(x^3\)[/tex].
- [tex]\(x^6\)[/tex] can be written as [tex]\((x^2)^3\)[/tex].
- [tex]\(x^3\)[/tex] can be written as [tex]\((x^1)^3\)[/tex].
- Therefore, this is not in the form of a sum of two distinct cubes.
5. [tex]\(27x^9 + x^{12}\)[/tex]
- The terms are [tex]\(27x^9\)[/tex] and [tex]\(x^{12}\)[/tex].
- [tex]\(27x^9\)[/tex] can be written as [tex]\((3x^3)^3\)[/tex].
- [tex]\(x^{12}\)[/tex] can be written as [tex]\((x^4)^3\)[/tex].
- Therefore, this is a sum of perfect cubes: [tex]\((3x^3)^3 + (x^4)^3\)[/tex].
6. [tex]\(9x^3 + 27x^9\)[/tex]
- The terms are [tex]\(9x^3\)[/tex] and [tex]\(27x^9\)[/tex].
- [tex]\(9x^3\)[/tex] can be written as [tex]\((3x)^3\)[/tex] because [tex]\(9 = 3^2\)[/tex] and [tex]\((x)^3\)[/tex].
- [tex]\(27x^9\)[/tex] can be written as [tex]\((3x^3)^3\)[/tex].
- Therefore, this is not in form of a sum of two distinct cubes.
The sums of perfect cubes in the given expressions are:
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(81x^3 + 16x^6\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
These expressions fit the form of sums of perfect cubes appropriately.
Now, let's examine each of the expressions provided:
1. [tex]\(8x^6 + 27\)[/tex]
- The terms are [tex]\(8x^6\)[/tex] and [tex]\(27\)[/tex].
- [tex]\(8x^6\)[/tex] can be written as [tex]\((2x^2)^3\)[/tex] because [tex]\(8 = 2^3\)[/tex] and [tex]\((x^2)^3 = x^6\)[/tex].
- [tex]\(27\)[/tex] is [tex]\(3^3\)[/tex].
- Therefore, this is a sum of perfect cubes: [tex]\((2x^2)^3 + 3^3\)[/tex].
2. [tex]\(x^9 + 1\)[/tex]
- The terms are [tex]\(x^9\)[/tex] and [tex]\(1\)[/tex].
- [tex]\(x^9\)[/tex] can be written as [tex]\((x^3)^3\)[/tex].
- [tex]\(1\)[/tex] is [tex]\(1^3\)[/tex].
- Therefore, this is a sum of perfect cubes: [tex]\((x^3)^3 + 1^3\)[/tex].
3. [tex]\(81x^3 + 16x^6\)[/tex]
- The terms are [tex]\(81x^3\)[/tex] and [tex]\(16x^6\)[/tex].
- [tex]\(81x^3\)[/tex] can be written as [tex]\((3x)^3\)[/tex] because [tex]\(81 = 3^4\)[/tex] and [tex]\(x^3\)[/tex].
- [tex]\(16x^6\)[/tex] can be written as [tex]\((2x^2)^3\)[/tex] because [tex]\(16 = 2^4\)[/tex].
- Therefore, this is a sum of perfect cubes: [tex]\((3x)^3 + (2x^2)^3\)[/tex].
4. [tex]\(x^6 + x^3\)[/tex]
- The terms are [tex]\(x^6\)[/tex] and [tex]\(x^3\)[/tex].
- [tex]\(x^6\)[/tex] can be written as [tex]\((x^2)^3\)[/tex].
- [tex]\(x^3\)[/tex] can be written as [tex]\((x^1)^3\)[/tex].
- Therefore, this is not in the form of a sum of two distinct cubes.
5. [tex]\(27x^9 + x^{12}\)[/tex]
- The terms are [tex]\(27x^9\)[/tex] and [tex]\(x^{12}\)[/tex].
- [tex]\(27x^9\)[/tex] can be written as [tex]\((3x^3)^3\)[/tex].
- [tex]\(x^{12}\)[/tex] can be written as [tex]\((x^4)^3\)[/tex].
- Therefore, this is a sum of perfect cubes: [tex]\((3x^3)^3 + (x^4)^3\)[/tex].
6. [tex]\(9x^3 + 27x^9\)[/tex]
- The terms are [tex]\(9x^3\)[/tex] and [tex]\(27x^9\)[/tex].
- [tex]\(9x^3\)[/tex] can be written as [tex]\((3x)^3\)[/tex] because [tex]\(9 = 3^2\)[/tex] and [tex]\((x)^3\)[/tex].
- [tex]\(27x^9\)[/tex] can be written as [tex]\((3x^3)^3\)[/tex].
- Therefore, this is not in form of a sum of two distinct cubes.
The sums of perfect cubes in the given expressions are:
- [tex]\(8x^6 + 27\)[/tex]
- [tex]\(x^9 + 1\)[/tex]
- [tex]\(81x^3 + 16x^6\)[/tex]
- [tex]\(27x^9 + x^{12}\)[/tex]
These expressions fit the form of sums of perfect cubes appropriately.
