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Answer :
We want to determine which expressions can be written as a sum of two perfect cubes. Recall that a perfect cube is any expression of the form [tex]$a^3$[/tex]. We check each option:
1. Expression:
[tex]$$x^9+1$$[/tex]
Notice that
[tex]$$x^9 = (x^3)^3 \quad \text{and} \quad 1 = 1^3.$$[/tex]
Thus, the expression is the sum of the cubes [tex]$(x^3)^3$[/tex] and [tex]$1^3$[/tex].
2. Expression:
[tex]$$8x^6+27$$[/tex]
Observe that
[tex]$$8x^6 = (2x^2)^3 \quad \text{and} \quad 27 = 3^3.$$[/tex]
Hence, this is the sum of the cubes [tex]$(2x^2)^3$[/tex] and [tex]$3^3$[/tex].
3. Expression:
[tex]$$x^6+x^3$$[/tex]
Here we can write
[tex]$$x^6 = (x^2)^3 \quad \text{and} \quad x^3 = x^3 = (x)^3.$$[/tex]
Therefore, it is the sum of [tex]$(x^2)^3$[/tex] and [tex]$(x)^3$[/tex].
4. Expression:
[tex]$$81x^3+16x^6$$[/tex]
In this case, [tex]$81x^3$[/tex] and [tex]$16x^6$[/tex] must both be perfect cubes. However,
- [tex]$81$[/tex] is not a perfect cube (since [tex]$3^3=27$[/tex] and [tex]$4^3=64$[/tex], etc.).
- [tex]$16$[/tex] is not a perfect cube either (as [tex]$2^3=8$[/tex] and [tex]$3^3=27$[/tex]).
Therefore, this expression is not a sum of two perfect cubes.
5. Expression:
[tex]$$27x^9+x^{12}$$[/tex]
Notice that
[tex]$$27x^9 = (3x^3)^3 \quad \text{and} \quad x^{12} = (x^4)^3.$$[/tex]
So, it can be written as the sum of the cubes [tex]$(3x^3)^3$[/tex] and [tex]$(x^4)^3$[/tex].
6. Expression:
[tex]$$9x^3+27x^9$$[/tex]
Although [tex]$27x^9$[/tex] is a cube (since [tex]$27x^9 = (3x^3)^3$[/tex]),
[tex]$$9x^3$$[/tex]
is not a perfect cube because 9 is not a cube number.
Therefore, this expression is not a sum of two perfect cubes.
Thus, the expressions that are sums of perfect cubes are:
- [tex]$x^9+1$[/tex]
- [tex]$8x^6+27$[/tex]
- [tex]$x^6+x^3$[/tex]
- [tex]$27x^9+x^{12}$[/tex]
The correct options are 1, 2, 3, and 5.
1. Expression:
[tex]$$x^9+1$$[/tex]
Notice that
[tex]$$x^9 = (x^3)^3 \quad \text{and} \quad 1 = 1^3.$$[/tex]
Thus, the expression is the sum of the cubes [tex]$(x^3)^3$[/tex] and [tex]$1^3$[/tex].
2. Expression:
[tex]$$8x^6+27$$[/tex]
Observe that
[tex]$$8x^6 = (2x^2)^3 \quad \text{and} \quad 27 = 3^3.$$[/tex]
Hence, this is the sum of the cubes [tex]$(2x^2)^3$[/tex] and [tex]$3^3$[/tex].
3. Expression:
[tex]$$x^6+x^3$$[/tex]
Here we can write
[tex]$$x^6 = (x^2)^3 \quad \text{and} \quad x^3 = x^3 = (x)^3.$$[/tex]
Therefore, it is the sum of [tex]$(x^2)^3$[/tex] and [tex]$(x)^3$[/tex].
4. Expression:
[tex]$$81x^3+16x^6$$[/tex]
In this case, [tex]$81x^3$[/tex] and [tex]$16x^6$[/tex] must both be perfect cubes. However,
- [tex]$81$[/tex] is not a perfect cube (since [tex]$3^3=27$[/tex] and [tex]$4^3=64$[/tex], etc.).
- [tex]$16$[/tex] is not a perfect cube either (as [tex]$2^3=8$[/tex] and [tex]$3^3=27$[/tex]).
Therefore, this expression is not a sum of two perfect cubes.
5. Expression:
[tex]$$27x^9+x^{12}$$[/tex]
Notice that
[tex]$$27x^9 = (3x^3)^3 \quad \text{and} \quad x^{12} = (x^4)^3.$$[/tex]
So, it can be written as the sum of the cubes [tex]$(3x^3)^3$[/tex] and [tex]$(x^4)^3$[/tex].
6. Expression:
[tex]$$9x^3+27x^9$$[/tex]
Although [tex]$27x^9$[/tex] is a cube (since [tex]$27x^9 = (3x^3)^3$[/tex]),
[tex]$$9x^3$$[/tex]
is not a perfect cube because 9 is not a cube number.
Therefore, this expression is not a sum of two perfect cubes.
Thus, the expressions that are sums of perfect cubes are:
- [tex]$x^9+1$[/tex]
- [tex]$8x^6+27$[/tex]
- [tex]$x^6+x^3$[/tex]
- [tex]$27x^9+x^{12}$[/tex]
The correct options are 1, 2, 3, and 5.
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