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Answer :
To determine whether an expression is a sum of perfect cubes, we need to check that each term can be written in the form
[tex]$$
(a\,x^k)^3 = a^3\,x^{3k}.
$$[/tex]
This means that for each term of the form [tex]$c\,x^p$[/tex]:
1. The exponent [tex]$p$[/tex] must be divisible by 3 (so that [tex]$p = 3k$[/tex] for some integer [tex]$k$[/tex]).
2. The coefficient [tex]$c$[/tex] must be a perfect cube (i.e. there must exist an integer [tex]$a$[/tex] such that [tex]$a^3 = c$[/tex]).
Let’s analyze each option:
1. Option 1: [tex]$\mathbf{8x^6+27}$[/tex]
- For the term [tex]$8x^6$[/tex]:
- The exponent [tex]$6$[/tex] is divisible by 3 since [tex]$6=3\cdot2$[/tex].
- The coefficient [tex]$8$[/tex] is a perfect cube because [tex]$2^3=8$[/tex].
- For the term [tex]$27$[/tex]:
- This is equivalent to [tex]$27x^0$[/tex] where the exponent [tex]$0$[/tex] is divisible by 3 ([tex]$0=3\cdot0$[/tex]).
- The coefficient [tex]$27$[/tex] is a perfect cube since [tex]$3^3=27$[/tex].
Since both terms are perfect cubes, the expression is a sum of perfect cubes.
2. Option 2: [tex]$\mathbf{x^9+1}$[/tex]
- For the term [tex]$x^9$[/tex]:
- The exponent [tex]$9$[/tex] is divisible by 3 ([tex]$9=3\cdot3$[/tex]).
- The coefficient is [tex]$1$[/tex], and [tex]$1^3=1$[/tex], so it is a perfect cube.
- For the term [tex]$1$[/tex]:
- This is [tex]$1x^0$[/tex]; the exponent [tex]$0$[/tex] is divisible by 3.
- The coefficient [tex]$1$[/tex] is a perfect cube since [tex]$1^3=1$[/tex].
Both terms are perfect cubes, so the expression is a sum of perfect cubes.
3. Option 3: [tex]$\mathbf{81x^3+16x^6}$[/tex]
- For the term [tex]$81x^3$[/tex]:
- The exponent [tex]$3$[/tex] is divisible by 3 ([tex]$3=3\cdot1$[/tex]).
- However, [tex]$81$[/tex] is not a perfect cube (since the closest cubes are [tex]$4^3=64$[/tex] and [tex]$5^3=125$[/tex]).
- For the term [tex]$16x^6$[/tex]:
- The exponent [tex]$6$[/tex] is divisible by 3 ([tex]$6=3\cdot2$[/tex]).
- The coefficient [tex]$16$[/tex] is not a perfect cube (as [tex]$2^3=8$[/tex] and [tex]$3^3=27$[/tex]).
Since at least one term is not a perfect cube, this expression is not a sum of perfect cubes.
4. Option 4: [tex]$\mathbf{x^6+x^3}$[/tex]
- For the term [tex]$x^6$[/tex]:
- The exponent [tex]$6$[/tex] is divisible by 3 ([tex]$6=3\cdot2$[/tex]).
- The coefficient is [tex]$1$[/tex], which is [tex]$1^3=1$[/tex].
- For the term [tex]$x^3$[/tex]:
- The exponent [tex]$3$[/tex] is divisible by 3 ([tex]$3=3\cdot1$[/tex]).
- The coefficient is also [tex]$1$[/tex], which is a perfect cube.
Thus, this expression is a sum of perfect cubes.
5. Option 5: [tex]$\mathbf{27x^9+x^{12}}$[/tex]
- For the term [tex]$27x^9$[/tex]:
- The exponent [tex]$9$[/tex] is divisible by 3 ([tex]$9=3\cdot3$[/tex]).
- The coefficient [tex]$27$[/tex] is a perfect cube since [tex]$3^3=27$[/tex].
- For the term [tex]$x^{12}$[/tex]:
- The exponent [tex]$12$[/tex] is divisible by 3 ([tex]$12=3\cdot4$[/tex]).
- The coefficient is [tex]$1$[/tex], which is [tex]$1^3=1$[/tex].
Therefore, this expression is a sum of perfect cubes.
6. Option 6: [tex]$\mathbf{9x^3+27x^9}$[/tex]
- For the term [tex]$9x^3$[/tex]:
- The exponent [tex]$3$[/tex] is divisible by 3 ([tex]$3=3\cdot1$[/tex]).
- The coefficient [tex]$9$[/tex] is not a perfect cube (since [tex]$2^3=8$[/tex] and [tex]$3^3=27$[/tex]).
- For the term [tex]$27x^9$[/tex]:
- The exponent [tex]$9$[/tex] is divisible by 3 ([tex]$9=3\cdot3$[/tex]).
- The coefficient [tex]$27$[/tex] is a perfect cube ([tex]$3^3=27$[/tex]).
Even though the second term is a perfect cube, the first term is not, so the expression is not a sum of perfect cubes.
After checking all options, the expressions that are sums of perfect cubes are those in options 1, 2, 4, and 5.
[tex]$$
(a\,x^k)^3 = a^3\,x^{3k}.
$$[/tex]
This means that for each term of the form [tex]$c\,x^p$[/tex]:
1. The exponent [tex]$p$[/tex] must be divisible by 3 (so that [tex]$p = 3k$[/tex] for some integer [tex]$k$[/tex]).
2. The coefficient [tex]$c$[/tex] must be a perfect cube (i.e. there must exist an integer [tex]$a$[/tex] such that [tex]$a^3 = c$[/tex]).
Let’s analyze each option:
1. Option 1: [tex]$\mathbf{8x^6+27}$[/tex]
- For the term [tex]$8x^6$[/tex]:
- The exponent [tex]$6$[/tex] is divisible by 3 since [tex]$6=3\cdot2$[/tex].
- The coefficient [tex]$8$[/tex] is a perfect cube because [tex]$2^3=8$[/tex].
- For the term [tex]$27$[/tex]:
- This is equivalent to [tex]$27x^0$[/tex] where the exponent [tex]$0$[/tex] is divisible by 3 ([tex]$0=3\cdot0$[/tex]).
- The coefficient [tex]$27$[/tex] is a perfect cube since [tex]$3^3=27$[/tex].
Since both terms are perfect cubes, the expression is a sum of perfect cubes.
2. Option 2: [tex]$\mathbf{x^9+1}$[/tex]
- For the term [tex]$x^9$[/tex]:
- The exponent [tex]$9$[/tex] is divisible by 3 ([tex]$9=3\cdot3$[/tex]).
- The coefficient is [tex]$1$[/tex], and [tex]$1^3=1$[/tex], so it is a perfect cube.
- For the term [tex]$1$[/tex]:
- This is [tex]$1x^0$[/tex]; the exponent [tex]$0$[/tex] is divisible by 3.
- The coefficient [tex]$1$[/tex] is a perfect cube since [tex]$1^3=1$[/tex].
Both terms are perfect cubes, so the expression is a sum of perfect cubes.
3. Option 3: [tex]$\mathbf{81x^3+16x^6}$[/tex]
- For the term [tex]$81x^3$[/tex]:
- The exponent [tex]$3$[/tex] is divisible by 3 ([tex]$3=3\cdot1$[/tex]).
- However, [tex]$81$[/tex] is not a perfect cube (since the closest cubes are [tex]$4^3=64$[/tex] and [tex]$5^3=125$[/tex]).
- For the term [tex]$16x^6$[/tex]:
- The exponent [tex]$6$[/tex] is divisible by 3 ([tex]$6=3\cdot2$[/tex]).
- The coefficient [tex]$16$[/tex] is not a perfect cube (as [tex]$2^3=8$[/tex] and [tex]$3^3=27$[/tex]).
Since at least one term is not a perfect cube, this expression is not a sum of perfect cubes.
4. Option 4: [tex]$\mathbf{x^6+x^3}$[/tex]
- For the term [tex]$x^6$[/tex]:
- The exponent [tex]$6$[/tex] is divisible by 3 ([tex]$6=3\cdot2$[/tex]).
- The coefficient is [tex]$1$[/tex], which is [tex]$1^3=1$[/tex].
- For the term [tex]$x^3$[/tex]:
- The exponent [tex]$3$[/tex] is divisible by 3 ([tex]$3=3\cdot1$[/tex]).
- The coefficient is also [tex]$1$[/tex], which is a perfect cube.
Thus, this expression is a sum of perfect cubes.
5. Option 5: [tex]$\mathbf{27x^9+x^{12}}$[/tex]
- For the term [tex]$27x^9$[/tex]:
- The exponent [tex]$9$[/tex] is divisible by 3 ([tex]$9=3\cdot3$[/tex]).
- The coefficient [tex]$27$[/tex] is a perfect cube since [tex]$3^3=27$[/tex].
- For the term [tex]$x^{12}$[/tex]:
- The exponent [tex]$12$[/tex] is divisible by 3 ([tex]$12=3\cdot4$[/tex]).
- The coefficient is [tex]$1$[/tex], which is [tex]$1^3=1$[/tex].
Therefore, this expression is a sum of perfect cubes.
6. Option 6: [tex]$\mathbf{9x^3+27x^9}$[/tex]
- For the term [tex]$9x^3$[/tex]:
- The exponent [tex]$3$[/tex] is divisible by 3 ([tex]$3=3\cdot1$[/tex]).
- The coefficient [tex]$9$[/tex] is not a perfect cube (since [tex]$2^3=8$[/tex] and [tex]$3^3=27$[/tex]).
- For the term [tex]$27x^9$[/tex]:
- The exponent [tex]$9$[/tex] is divisible by 3 ([tex]$9=3\cdot3$[/tex]).
- The coefficient [tex]$27$[/tex] is a perfect cube ([tex]$3^3=27$[/tex]).
Even though the second term is a perfect cube, the first term is not, so the expression is not a sum of perfect cubes.
After checking all options, the expressions that are sums of perfect cubes are those in options 1, 2, 4, and 5.
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