Final answer:
The expression that is a sum of cubes is 27x⁹y⁶
Step-by-step explanation:
The expression that is a sum of cubes is 4) 27x⁹y⁶. To determine if an expression is a sum of cubes, we need to find two terms raised to the power of 3 that add up to the given expression. In this case, (3x³) + (3y²) = 27x⁹y⁶, which satisfies the criteria for a sum of cubes.
Let's examine each option provided:
-24a⁵ isn't a sum of cubes since the exponent 5 is not a multiple of 3.
125bⁱ⁸ seems like a sum of cubes because the number 125 is 5 cubed (5³), but the exponent of b is 18, which is also divisible by 3 (b¶³). Hence, this can be seen as (5b⁶)³.
-6a²⁷b⁸ isn't a sum of cubes as the exponents 27 and 8 are not divisible by 3 for a and b respectively.
27x⁹y⁶ is definitely a sum of cubes, where 27 is equal to 3 cubed (3³) and both x and y are raised to a power that is a multiple of 3 (x³⁹ and y³²).
Among the provided options, 27x⁹y⁶ and 125bⁱ⁸ are both sums of cubes where the entire expression can be represented as a cube of something else. Specifically, 27x⁹y⁶ can be rewritten as (3x³y²)³ and 125bⁱ⁸ as (5b⁶)³ which fits the criteria for a sum of cubes.