Can someone please help me with this?

Question

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

There may be more than one correct answer. Be sure to select all that apply.

 

8m^5-12n^3

p^15+8q^12

64a^6-343b^12

64y^3+27y^6

Answer 1

You want to look if the powers are multiples of 3 to help you look for cube terms.

Answer 2

To be a sum or difference of cubes all the terms have to have exponents that can be expressed as a multiple of 3.

So for instance X^4 can not be expressed as a cube, but X^6 can X^6=(X^3)^2= X^(3*2).  

 

Hope this gets you on the right track.

 

Mark 

Answer 3

Alright, Mari, let's look at what the difference of cubes means.  To cube something means that you multiply it by itself three times, like 5^3 =5 x 5 x 5 = 125, so the cube root of 125 is 5.  Do any of the above fit that with both constants?

 

Now, also remember that b^3 x b^3 x b^3 = b^(3+3+3) = b^9, so the cube root of b^9 is b^3.  Do any of the above fit this with both sets of variables?

 

I see three of them that work. Can you find them?

Answer 4

There are 3 answers:

 

Remember, sum of cube in factor form is: (a + b) (a^2 + ab + b^2)

          difference of cube in factor form is: (a - b) (a^2 - ab + b^2)

 

p^15 + 8q^12 is a sum of cubes.  If you take cube root for p^15, it will give you P^5 and if you take cube root of 8q^12, it will give you 2q^4.  This can be factored as (p^5 + 2q^4) (p^10 - 2p^5q^4 + 4q^8).

 

64a^6 - 343b^12 is a difference of cubes.  If you take cube root for 64a^6, it will give you 4a^2 and if you take cube root of -343b^12, it will give you -7b^4.  This can be factor as (4a^2 - 7b^4) (16a^4 + 28a^2b^4 + 49b^8).

 

64y^3 + 27y^6 is a sum of cubes.  It will finally factor as (y^3) (4 + 3y) (4 - 3y)^2

                                                                                   (y^3) (4 + 3y) (4 - 3y) (4 - 3y).

 

You can factor out the GCF (Greatest Common Factor: y^3) out first and still have a sum of cubes: 64 + 27y^3.

Or you can let a = 4y and b = 3y^2 to start.  Either way, it will come down to same final factor form at the end.  Try it.  Good luck to you.

 

Looks like a lot of people missed that (7)^3 is 343.  Which means (-7)^3 is -343.

 

First one is neither a sum of cubes nor a difference of cubes because of exponent is 5 (not multiple of 3).

Answer 5

Each expression has two terms. They are either added or subtracted which is why they are considered a sum or difference. If a term is a cube, that means it has a cube root.

 

You need to determine whether each term has a perfect cube root.

When you does this, consider the number separately from the variable.

 

I'll do the first one for you.

 

8m^5-12n^3

 

Start with the first term 8m^5. The number has a cube root. The cube root of 8 is 2 because 2 x 2 x 2 = 8

m^5 does not have a cube root because m raised to a power multiplied by itself 3 times must equal m^5 and there is no such power that is an integer.

 

let me show you.

If m^a x m^a x m^a = m⁵

then m^3a = m⁵

3a = 5

a = 5/3 which is not an integer

 

Although we already know the first expression is not a difference of cubes because the first term is not a cube, let's check the second term for practice.

 

12n^3

 

The cube root of n³ is n because n x  n x n = n³

However, the number 12 is not a perfect cube. The cube root of 12 is not an integer.

 

Remember, when you are checking to see if the variable parts are perfect cubes, the rule to know is this,

 

When you multiply powers with the same base, you get a single term with the same base and you add the exponents.

 

It looks like this.

 

(X^a)(X^b) = X^a+b

 

So for the second expression p^15+8q^12

 

p⁵p⁵p⁵ = p¹⁵     2 x 2 x 2 = 8     q⁴q⁴q⁴ = q¹²

 

Now you should be able to work out the others