Question

B. Write S if the expression is a sum of two cubes, D if a difference of two cubes, and N if neither.

1. 27x⁶ + y³=
2. 81 - b¹⁵=
3. 64x³ - 9z⁹=
4. 36m¹² - n⁶
5. 1 - d¹²=
6. 729 - y²⁹=
7. 343 - y⁶=
8. 4x³ + 8 =
9. 144 -125y² =
10. 64j⁶- k⁹=​​​

Answer 1

Answers:

1. S
2. N
3. N
4. N
5. D
6. N
7. D
8. N
9. N
10. D

==========================================================

Step-by-step explanation:

Question 1

27x^6 = (3x^2)^3 which is one cube and y^3 is another cube

Therefore 27x^6+y^3 is a sum of two cubes.

It might help to think of it like A^3+B^3 where in this case A = 3x^2 and B = y.

----------------------------------------

Question 2

81 isn't a perfect cube since 81^(1/3) = 4.3267 approximately, which isn't a whole number. We need 81^(1/3) to be a whole number if we wanted 81 to be a perfect cube.

We don't even need to check b^15 since 81 being a non-perfect cube means the entire expression cannot be a sum of two cubes, nor a difference of cubes.

----------------------------------------

Question 3

64x^3 = (4x)^3 is a perfect cube

but 9z^9 is not a perfect cube

We can see this if we computed 9^(1/3) and the result is a non-whole number.

The answer here is the same as the previous question.

----------------------------------------

Question 4

36^(1/3) is not a whole number, so 36 isn't a perfect cube. By extension 36m^12 isn't a perfect cube either. The answer is the same as questions 2 and 3.

----------------------------------------

Question 5

1 is a perfect cube since 1 = 1^3

d^12 is a perfect cube because d^12 = (d^4)^3

Therefore, we have a difference of two cubes.

----------------------------------------

Question 6

729^(1/3) = 9 which rearranges to 9^3 = 729, showing 729 is a perfect cube.

However y^29 is not a perfect cube since the exponent 29 is not a multiple of 3.

The overall expression is "neither".

----------------------------------------

Question 7

343^(1/3) = 7 rearranges to 7^3 = 343, showing 343 is a perfect cube

y^6 is a perfect cube since (y^2)^3, i.e. the exponent 6 is a multiple of 3.

We have a difference of cubes.

----------------------------------------

Question 8

4 isn't a perfect cube since 4^(1/3) results in some decimal value that isn't a whole number. The entire expression is neither a sum nor difference of cubes.

----------------------------------------

Question 9

144^(1/3) isn't a whole number, so we get a similar result to problem 8.

----------------------------------------

Question 10

64j^6 = (4j^3)^3 is one cube

k^9 = (k^3)^3 is another cube

We have a difference of cubes