Difference of Cubes? (a^3 - b^3) answer to it please.

[justin]

Difference of Cubes? (a^3 - b^3)

also (a^4 - b^4)

what do these equal

 

[pka]

What about these?

 

[stapel]

Re: Difference of Cubes? (a^3 - b^3)

Difference of Cubes? (a^3 - b^3)

Yes, "a³ - b³" is a difference of cubes.

also (a^4 - b^4)

This would be a difference of fourth powers. (There may be another name for "to the fourth power", but this terminology should be fine.)

Eliz.

 

[TchrWill]

Difference of Cubes? (a^3 - b^3)

also (a^4 - b^4)

what do these equal

They could be set equal to anything basically. For instance:

Find two perfect cubes whose difference is a perfect square.

Letting x = a, y = ma (m > 1) and z = na, we can write a^3(m^3 - 1) = n^2(a^2) or a = n^2/(m^3 - 1).
We need to find compatible integer values of "m" and "n" that produce integer values for "a".
For a given "m", search for values of "n" that result in n^2 being evenly divisible by (m^3 - 1) producing "a".
Alternatively, for m = 2 on up, a = n^2/7 for m = 2, = n^2/26 for m = 3, = n^2/63 for m = 4, etc.

m......n--->...1.....2.....3.....4.....5.....6.....7.....8.....9.....10.....11.....12.....13.....14.....15
\/
1...............................................................................................................................none

2.........................................................7....................................................28.............from n = 7x1, 2, 3,...i

3...............................................................................................................................from n = 26x1, 2, 3,...i

Values of "n" for higher values of "m" must be derived in the same manner as those given above.

Check: For n = 7 and m = 2, a = 7^2/(8 - 1) = 7 making x = 7, y = 14 and z = 49.
Then, 14^3 - 7^3 = 49^2 or 2744 - 343 = 2401.


y^4 - x^4 = z^3

FInd two numbers, the difference of whose fourth powers is a perfect cube.

Let x = a, y = ma and z = na.
Then, m^4a^4 - a^4 = n^3a^3 or a = n^3/(m^4 - 1).
By inspection, m must be greater than 1 and therefore, m = 1 and n = an even number cannot lead to integer answers for x, y and z.
With m > 1, all answers are rational with no integer answers being possible.

Example:

Letting m = 2 and n = 5 makes a = 125/15.
Then, x = a = 125/17, y = 250/17 and z = 625/17.
Then, (250/15)^4 - (125/15)^4 = (625/15)^3.

 

[stapel]

Note: When posting replies and clarifications, it is best to post them as replies, rather than as "stealth edits", which are frequently overlooked and can lead to confusion.

what do these equal

You have clarified your original post somewhat: You appear now to be asking regarding the values of these two expressions. But until you provide the values for "a" and "b", these expressions cannot be evaluated.

If, perhaps, the instructions were actually to "expand the following expressions fully" or some other meaning that we haven't guessed yet, please reply with this clarification, showing the steps you have tried thus far, so we can see where you are needing help.

Thank you.

Eliz.