Answer :
We want to determine which expressions are sums of perfect cubes. An expression is a sum of perfect cubes if each term can be written in the form [tex]$a^3$[/tex], where [tex]$a$[/tex] is an expression.
Below is the step-by-step check for each option:
1. [tex]$$8x^6 + 27$$[/tex]
- Notice that
[tex]$$8x^6 = (2x^2)^3$$[/tex]
because
[tex]$$(2x^2)^3 = 2^3 \cdot (x^2)^3 = 8x^6.$$[/tex]
- Also,
[tex]$$27 = 3^3.$$[/tex]
- Since both terms are perfect cubes, this expression is a sum of perfect cubes.
2. [tex]$$x^9 + 1$$[/tex]
- We can write
[tex]$$x^9 = (x^3)^3$$[/tex]
because
[tex]$$(x^3)^3 = x^9.$$[/tex]
- Also,
[tex]$$1 = 1^3.$$[/tex]
- Thus, this expression is also a sum of perfect cubes.
3. [tex]$$81x^3 + 16x^6$$[/tex]
- The coefficient [tex]$81$[/tex] is not a perfect cube (since [tex]$3^3 = 27$[/tex] and [tex]$4^3 = 64$[/tex], neither of which is [tex]$81$[/tex]).
- Similarly, the coefficient [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]
- We have
[tex]$$x^6 = (x^2)^3$$[/tex]
because
[tex]$$(x^2)^3 = x^6.$$[/tex]
- And,
[tex]$$x^3 = (x)^3.$$[/tex]
- Since both terms are perfect cubes, this expression is a sum of perfect cubes.
5. [tex]$$27x^9 + x^{12}$$[/tex]
- For the first term:
[tex]$$27x^9 = (3x^3)^3$$[/tex]
because
[tex]$$(3x^3)^3 = 27x^9.$$[/tex]
- For the second term:
[tex]$$x^{12} = (x^4)^3$$[/tex]
since
[tex]$$(x^4)^3 = x^{12}.$$[/tex]
- Hence, this expression is a sum of perfect cubes.
6. [tex]$$9x^3 + 27x^9$$[/tex]
- Here, while
[tex]$$27x^9 = (3x^3)^3,$$[/tex]
the term [tex]$$9x^3$$[/tex] does not represent a perfect cube because [tex]$9$[/tex] is not a perfect cube (its cube root is not an integer).
- Thus, this expression is not a sum of perfect cubes.
Based on these computations, the expressions that are sums of perfect cubes are options 1, 2, 4, and 5.
Below is the step-by-step check for each option:
1. [tex]$$8x^6 + 27$$[/tex]
- Notice that
[tex]$$8x^6 = (2x^2)^3$$[/tex]
because
[tex]$$(2x^2)^3 = 2^3 \cdot (x^2)^3 = 8x^6.$$[/tex]
- Also,
[tex]$$27 = 3^3.$$[/tex]
- Since both terms are perfect cubes, this expression is a sum of perfect cubes.
2. [tex]$$x^9 + 1$$[/tex]
- We can write
[tex]$$x^9 = (x^3)^3$$[/tex]
because
[tex]$$(x^3)^3 = x^9.$$[/tex]
- Also,
[tex]$$1 = 1^3.$$[/tex]
- Thus, this expression is also a sum of perfect cubes.
3. [tex]$$81x^3 + 16x^6$$[/tex]
- The coefficient [tex]$81$[/tex] is not a perfect cube (since [tex]$3^3 = 27$[/tex] and [tex]$4^3 = 64$[/tex], neither of which is [tex]$81$[/tex]).
- Similarly, the coefficient [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]
- We have
[tex]$$x^6 = (x^2)^3$$[/tex]
because
[tex]$$(x^2)^3 = x^6.$$[/tex]
- And,
[tex]$$x^3 = (x)^3.$$[/tex]
- Since both terms are perfect cubes, this expression is a sum of perfect cubes.
5. [tex]$$27x^9 + x^{12}$$[/tex]
- For the first term:
[tex]$$27x^9 = (3x^3)^3$$[/tex]
because
[tex]$$(3x^3)^3 = 27x^9.$$[/tex]
- For the second term:
[tex]$$x^{12} = (x^4)^3$$[/tex]
since
[tex]$$(x^4)^3 = x^{12}.$$[/tex]
- Hence, this expression is a sum of perfect cubes.
6. [tex]$$9x^3 + 27x^9$$[/tex]
- Here, while
[tex]$$27x^9 = (3x^3)^3,$$[/tex]
the term [tex]$$9x^3$$[/tex] does not represent a perfect cube because [tex]$9$[/tex] is not a perfect cube (its cube root is not an integer).
- Thus, this expression is not a sum of perfect cubes.
Based on these computations, the expressions that are sums of perfect cubes are options 1, 2, 4, and 5.
