which expression is a sum of cubes?\n-27a³b⁶ + 8a⁹b¹²\n-9a³b⁶ + a⁹b¹⁰\n9a³b⁶ + 8a⁹b¹²\n27a³b⁸ + 8a⁹b¹²
which expression is a sum of cubes?\n-27a³b⁶ + 8a⁹b¹²\n-9a³b⁶ + a⁹b¹⁰\n9a³b⁶ + 8a⁹b¹²\n27a³b⁸ + 8a⁹b¹²
Answer
Explanation:
Step1: Recall the form of sum of cubes
The sum of cubes formula is (x^{3}+y^{3}), where (x) and (y) are expressions. For a term (Ax^{m}y^{n}) to be a perfect - cube term, (A) must be a perfect cube ((A = k^{3},k\in\mathbb{Z})), and (m) and (n) must be multiples of (3).
Step2: Analyze each option
- Option 1: (-27a^{3}b^{6}+8a^{9}b^{12})
- For (-27a^{3}b^{6}), (-27=(- 3)^{3}), (a^{3}=(a)^{3}), (b^{6}=(b^{2})^{3}). So, (-27a^{3}b^{6}=(-3ab^{2})^{3}).
- For (8a^{9}b^{12}), (8 = 2^{3}), (a^{9}=(a^{3})^{3}), (b^{12}=(b^{4})^{3}). So, (8a^{9}b^{12}=(2a^{3}b^{4})^{3}).
- The expression (-27a^{3}b^{6}+8a^{9}b^{12}) is of the form (x^{3}+y^{3}) (where (x=-3ab^{2}) and (y = 2a^{3}b^{4})).
- Option 2: (-9a^{3}b^{6}+a^{9}b^{10})
- Since (-9) is not a perfect cube ((\sqrt[3]{-9}\notin\mathbb{Z})) and (10) is not a multiple of (3), this is not a sum of cubes.
- Option 3: (9a^{3}b^{6}+8a^{9}b^{12})
- Since (9) is not a perfect cube ((\sqrt[3]{9}\notin\mathbb{Z})), this is not a sum of cubes.
- Option 4: (27a^{3}b^{8}+8a^{9}b^{12})
- Since (8) is a perfect cube ((8 = 2^{3})), (a^{9}=(a^{3})^{3}), (b^{12}=(b^{4})^{3}), but (b^{8}) (because (8) is not a multiple of (3)) makes (27a^{3}b^{8}) not a perfect - cube term ((27a^{3}b^{8}=3^{3}a^{3}b^{8}), and (b^{8}) is not a cube of an integer - power of (b)).
Answer:
(-27a^{3}b^{6}+8a^{9}b^{12})