Question

Which are sums of perfect cubes? Check all that apply.

A=8x6 + 27
B=x9 + 1
C=81x3 + 16x6
D=x6 + x3
E=27x9 + x12
F=9x3 + 27x9

Answer 1

The sums of perfect cubes are options A, C, E, and F.

Step-by-step explanation:

The sums of perfect cubes can be found by cubing the digit term in the usual way and multiplying the exponent of the exponential term by 3. Let's check which options from A to F are sums of perfect cubes:

• Option A: 8x6 + 27 = (2^3)x^6 + 3^3 = 2^3x^6 + 3^3 = (2x^2 + 3)^3, which is a perfect cube.
• Option B: x^9 + 1 = (x^3)^3 + 1 = (x^3 + 1)(x^6 - x^3 + 1), which is not a perfect cube.
• Option C: 81x^3 + 16x^6 = (3^4)(x^1)^3 + (2^4)(x^2)^3 = 3^4x^1^3 + 2^4x^2^3 = (3x + 2x^2)^3, which is a perfect cube.
• Option D: x^6 + x^3 = (x^2)^3 + (x)^3 = (x^2 + x)(x^4 - x^3 + x^2), which is not a perfect cube.
• Option E: 27x^9 + x^12 = (3^3)(x^3)^3 + (x^4)^3 = 3^3x^3^3 + x^4^3 = (3x^3 + x^4)^3, which is a perfect cube.
• Option F: 9x^3 + 27x^9 = (3^2)(x^1)^3 + (3^3)(x^3)^3 = 3^2x^1^3 + 3^3x^3^3 = (3x + 3x^3)^3, which is a perfect cube.

Therefore, the sums of perfect cubes are options A, C, E, and F.

Answer 2

I think these are the sums of perfect cubes.

A = (2x²)³ + (3)³
B = (x³)³ + (1)³
D = (x²)³ + (x)³
E = (3x³)³ + (x^4)³