Question

Which of the expressions below is a sum or difference of cubes?

8m^5 – 12n^3

64y^3 + 27y^6

64a^6 – 343b^12

p^15 + 8q^12
Can someone help me? Im not sure how to figure this out.

Answer 1

You need the two terms to be cubes of something.

Answer 2

For a term to be a perfect cube of something, both the number part and the variable part have to be perfect cubes.

Answer 3

Let's look at choice 1.

8m^5 and 12n^3

8 is the cube of 2
m^5 is not a perfect5 cube
That means the first term is not a perfect cube.

12 is not a perfect cube
n^3 is a perfect cube
That means the term 12bn^3 is not a perfect cube.

Since parts of both terms are not perfect cubes, choice 1 is not a difference of cubes.

Answer 4

Did you understand the explanation above?

Answer 5

A little?

Answer 6

Let's try to change that to "a lot."
The perfect cube numbers are the numbers that when you take the cubic root, you get a rational number.
To come up with perfect cubes, try cubing whole numbers:
0^3 = 0
1^3 = 1
2^3 = 8
3^3 = 27
4^3 = 64
5^3 = 125
From now on, when you see the numbers 0, 1, 8, 27, 64, 125, you should remember that they are perfect cubes. This is simply memorization.
Of course, there are many more perfect cubes (it is an infinite list) but these are the numbers you will see most often in algebra problems.

Answer 7

Once again, keep the numbers 1, 8, 27, 64, 125 in mind as being perfect cubes.
Let's make our list a little longer up to 10^3.
6^3 = 216
7^3 = 343
8^3 = 512
9^3 = 729
10^3 = 1000

Answer 8

Ok, keep these numbers in mind as being perfect cubes:
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Answer 9

Any term that has one of those numbers is possibly a perfect cube.
The next thing a term needs to have to be a perfect cube is a variable part that is also a perfect cube.
The variable part is the part that has variables (letters).
For a variable to be a perfect cube, it has to have an exponent that is a multiple of 3.

Answer 10

For example, x^3, y^6, w^9, x^12 are all perfect cubes.
x^8, y^4, w^10 are not perfect cubes.

Answer 11

So far so good?

Answer 12

Now look for expressions in your choices in which all numbers and all variables are perfect cubes according to my explanations above. Those are the ones that are the sums or subtractions of cubes.