Answer :
We need to check if each expression is a sum of perfect cubes. A perfect cube is any number (or expression) that can be written in the form [tex]$a^3$[/tex], where [tex]$a$[/tex] is an integer (or an expression in terms of the variable).
Let's analyze each expression:
1. [tex]$$8x^6 + 27$$[/tex]
- Notice that [tex]$$8x^6 = (2x^2)^3$$[/tex] because [tex]$$8 = 2^3$$[/tex] and [tex]$$x^6 = (x^2)^3.$$[/tex]
- Also, [tex]$$27 = 3^3.$$[/tex]
Since both terms are perfect cubes, the expression is a sum of perfect cubes.
2. [tex]$$x^9 + 1$$[/tex]
- Write [tex]$$x^9 = (x^3)^3.$$[/tex]
- And [tex]$$1 = 1^3.$$[/tex]
So this expression is also a sum of perfect cubes.
3. [tex]$$81x^3 + 16x^6$$[/tex]
- The number [tex]$$81$$[/tex] is not a perfect cube (since [tex]$$4^3 = 64$$[/tex] and [tex]$$5^3 = 125$$[/tex], [tex]$$81$$[/tex] lies between them).
- Also, [tex]$$16$$[/tex] is not a perfect cube.
Therefore, this expression is not a sum of perfect cubes.
4. [tex]$$x^6 + x^3$$[/tex]
- Write [tex]$$x^6 = (x^2)^3.$$[/tex]
- And [tex]$$x^3 = (x)^3.$$[/tex]
Both terms are perfect cubes, so this is a sum of perfect cubes.
5. [tex]$$27x^9 + x^{12}$$[/tex]
- Here, [tex]$$27 = 3^3$$[/tex] and [tex]$$x^9 = (x^3)^3.$$[/tex]
- Also, [tex]$$x^{12} = (x^4)^3.$$[/tex]
Thus, each term is a perfect cube, and the expression is a sum of perfect cubes.
6. [tex]$$9x^3 + 27x^9$$[/tex]
- Although [tex]$$27x^9$$[/tex] can be written as [tex]$$(3x^3)^3,$$[/tex] the number [tex]$$9$$[/tex] is not a perfect cube since there is no integer [tex]$a$[/tex] such that [tex]$$a^3 = 9.$$[/tex]
So this expression is not a sum of perfect cubes.
After our analysis, the expressions that are sums of perfect cubes are numbers 1, 2, 4, and 5.
Let's analyze each expression:
1. [tex]$$8x^6 + 27$$[/tex]
- Notice that [tex]$$8x^6 = (2x^2)^3$$[/tex] because [tex]$$8 = 2^3$$[/tex] and [tex]$$x^6 = (x^2)^3.$$[/tex]
- Also, [tex]$$27 = 3^3.$$[/tex]
Since both terms are perfect cubes, the expression is a sum of perfect cubes.
2. [tex]$$x^9 + 1$$[/tex]
- Write [tex]$$x^9 = (x^3)^3.$$[/tex]
- And [tex]$$1 = 1^3.$$[/tex]
So this expression is also a sum of perfect cubes.
3. [tex]$$81x^3 + 16x^6$$[/tex]
- The number [tex]$$81$$[/tex] is not a perfect cube (since [tex]$$4^3 = 64$$[/tex] and [tex]$$5^3 = 125$$[/tex], [tex]$$81$$[/tex] lies between them).
- Also, [tex]$$16$$[/tex] is not a perfect cube.
Therefore, this expression is not a sum of perfect cubes.
4. [tex]$$x^6 + x^3$$[/tex]
- Write [tex]$$x^6 = (x^2)^3.$$[/tex]
- And [tex]$$x^3 = (x)^3.$$[/tex]
Both terms are perfect cubes, so this is a sum of perfect cubes.
5. [tex]$$27x^9 + x^{12}$$[/tex]
- Here, [tex]$$27 = 3^3$$[/tex] and [tex]$$x^9 = (x^3)^3.$$[/tex]
- Also, [tex]$$x^{12} = (x^4)^3.$$[/tex]
Thus, each term is a perfect cube, and the expression is a sum of perfect cubes.
6. [tex]$$9x^3 + 27x^9$$[/tex]
- Although [tex]$$27x^9$$[/tex] can be written as [tex]$$(3x^3)^3,$$[/tex] the number [tex]$$9$$[/tex] is not a perfect cube since there is no integer [tex]$a$[/tex] such that [tex]$$a^3 = 9.$$[/tex]
So this expression is not a sum of perfect cubes.
After our analysis, the expressions that are sums of perfect cubes are numbers 1, 2, 4, and 5.
