Sum of Cubes

[ErikaG895]

Source: Manhattan GMAT CAT Exam Review Question for Quant.

If the sum of the cubes of a and b is 8 and a^6 - b^6 = 14, what is the value of a^3?

The explanation says:

We know that the sum of the cubes of a and b is 8: a^3 + b^3 = 8. We also know that a^6 – b^6 = 14. Using our knowledge of the quadratic template for the difference of two squares,
x2 – y2 = (x + y)(x – y), we can rewrite a^6 – b^6 = 14 as follows:

I undertstand the above

(a^3)2 – (b^3)2 = 14
(a^3 – b3)(a^3 + b^3) = 14

I am not sure how or even why the math above is done.

[jlucero]

Source: Manhattan GMAT CAT Exam Review Question for Quant.

If the sum of the cubes of a and b is 8 and a^6 - b^6 = 14, what is the value of a^3?

The explanation says:

We know that the sum of the cubes of a and b is 8: a^3 + b^3 = 8. We also know that a^6 – b^6 = 14. Using our knowledge of the quadratic template for the difference of two squares,
x2 – y2 = (x + y)(x – y), we can rewrite a^6 – b^6 = 14 as follows:

I undertstand the above

(a^3)2 – (b^3)2 = 14
(a^3 – b3)(a^3 + b^3) = 14

I am not sure how or even why the math above is done.

Hopefully you remember how to foil out the expression:
(a^3 – b3)(a^3 + b^3)
F = a^3 * a^3
O = a^3 * b^3
I = -b^3 * a^3
L = = -b^3 * b^3

Notice that the O and I terms will cancel out when you add them together, leaving

(a^3)^2 - (b^3)^2
a^6 - b^6

This is a special product that is best to simply memorize, since it shows up a lot on this test. The three to memorize are the top three on this image:

http://concepts.ck12.org/preview/specia ... omials.jpg

[RonPurewal]

When you learn formulas, make sure you learn to see them as relationships.

E.g., this whole a^2 – b^2 = (a – b)(a + b) thing.
It's not important that the things are called "a" and "b". Nor are any of the other superficial features important.
What's important is that the things on the left are the squares of the things on the right.
Or, alternatively, the things on the right are the square roots of the things on the left.

If you realize that this is a general relationship, then it should be easy to see that a^6 – b^6 = (a^3 – b^3)(a^3 + b^3), because squaring something^3 gives something^6 (or, alternatively, taking the square root of something^6 gives something^3).

Similarly, a – b = (√a – √b)(√a + √b). Same relationship.

Etc.