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Answer :
We want to check which expressions can be written in the form
[tex]$$
a^3 + b^3,
$$[/tex]
where both [tex]$a$[/tex] and [tex]$b$[/tex] are expressions (involving [tex]$x$[/tex] or numbers) and their cubes give the terms in the sum.
Let’s check each option:
1. [tex]$$8x^6 + 27$$[/tex]
Notice that
[tex]$$8x^6 = (2x^2)^3$$[/tex]
and
[tex]$$27 = 3^3.$$[/tex]
Therefore,
[tex]$$8x^6 + 27 = (2x^2)^3 + 3^3,$$[/tex]
which is a sum of perfect cubes.
2. [tex]$$x^9 + 1$$[/tex]
Observe that
[tex]$$x^9 = (x^3)^3$$[/tex]
and also
[tex]$$1 = 1^3.$$[/tex]
Thus,
[tex]$$x^9 + 1 = (x^3)^3 + 1^3,$$[/tex]
so this is also a sum of perfect cubes.
3. [tex]$$81x^3 + 16x^6$$[/tex]
For an expression to be a perfect cube, its coefficient must be a perfect cube. Here, [tex]$81$[/tex] and [tex]$16$[/tex] are not perfect cubes (since [tex]$\sqrt[3]{81}$[/tex] and [tex]$\sqrt[3]{16}$[/tex] are not integers). Therefore, this expression cannot be expressed as a sum of perfect cubes.
4. [tex]$$x^6 + x^3$$[/tex]
We can write
[tex]$$x^6 = (x^2)^3$$[/tex]
and
[tex]$$x^3 = x^3 = (x)^3.$$[/tex]
Hence,
[tex]$$x^6 + x^3 = (x^2)^3 + (x)^3,$$[/tex]
which is a sum of perfect cubes.
5. [tex]$$27x^9 + x^{12}$$[/tex]
Notice that
[tex]$$27x^9 = (3x^3)^3$$[/tex]
since [tex]$27 = 3^3$[/tex], and
[tex]$$x^{12} = (x^4)^3.$$[/tex]
Thus,
[tex]$$27x^9 + x^{12} = (3x^3)^3 + (x^4)^3,$$[/tex]
making it a sum of perfect cubes.
6. [tex]$$9x^3 + 27x^9$$[/tex]
While [tex]$27x^9$[/tex] can be written as [tex]$(3x^3)^3$[/tex], the coefficient [tex]$9$[/tex] in [tex]$9x^3$[/tex] is not a perfect cube (because [tex]$\sqrt[3]{9}$[/tex] is not an integer) and therefore it cannot be written in the form of a cube of a simpler expression. Thus, this is not a sum of perfect cubes.
In summary, the expressions that are sums of perfect cubes are found in options 1, 2, 4, and 5.
[tex]$$
a^3 + b^3,
$$[/tex]
where both [tex]$a$[/tex] and [tex]$b$[/tex] are expressions (involving [tex]$x$[/tex] or numbers) and their cubes give the terms in the sum.
Let’s check each option:
1. [tex]$$8x^6 + 27$$[/tex]
Notice that
[tex]$$8x^6 = (2x^2)^3$$[/tex]
and
[tex]$$27 = 3^3.$$[/tex]
Therefore,
[tex]$$8x^6 + 27 = (2x^2)^3 + 3^3,$$[/tex]
which is a sum of perfect cubes.
2. [tex]$$x^9 + 1$$[/tex]
Observe that
[tex]$$x^9 = (x^3)^3$$[/tex]
and also
[tex]$$1 = 1^3.$$[/tex]
Thus,
[tex]$$x^9 + 1 = (x^3)^3 + 1^3,$$[/tex]
so this is also a sum of perfect cubes.
3. [tex]$$81x^3 + 16x^6$$[/tex]
For an expression to be a perfect cube, its coefficient must be a perfect cube. Here, [tex]$81$[/tex] and [tex]$16$[/tex] are not perfect cubes (since [tex]$\sqrt[3]{81}$[/tex] and [tex]$\sqrt[3]{16}$[/tex] are not integers). Therefore, this expression cannot be expressed as a sum of perfect cubes.
4. [tex]$$x^6 + x^3$$[/tex]
We can write
[tex]$$x^6 = (x^2)^3$$[/tex]
and
[tex]$$x^3 = x^3 = (x)^3.$$[/tex]
Hence,
[tex]$$x^6 + x^3 = (x^2)^3 + (x)^3,$$[/tex]
which is a sum of perfect cubes.
5. [tex]$$27x^9 + x^{12}$$[/tex]
Notice that
[tex]$$27x^9 = (3x^3)^3$$[/tex]
since [tex]$27 = 3^3$[/tex], and
[tex]$$x^{12} = (x^4)^3.$$[/tex]
Thus,
[tex]$$27x^9 + x^{12} = (3x^3)^3 + (x^4)^3,$$[/tex]
making it a sum of perfect cubes.
6. [tex]$$9x^3 + 27x^9$$[/tex]
While [tex]$27x^9$[/tex] can be written as [tex]$(3x^3)^3$[/tex], the coefficient [tex]$9$[/tex] in [tex]$9x^3$[/tex] is not a perfect cube (because [tex]$\sqrt[3]{9}$[/tex] is not an integer) and therefore it cannot be written in the form of a cube of a simpler expression. Thus, this is not a sum of perfect cubes.
In summary, the expressions that are sums of perfect cubes are found in options 1, 2, 4, and 5.
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