from GMAT Prep; a^4-b^4

[xander1922]

Can someone walk me through this one?

What is the value of a^4-b^4?

1) a^2-b^2=16

2) a+b=8

Thanks!

[leovir.22]

Hi,

Is the answer option c)?

[JonathanSchneider]

The first thing you want to do is recognize that we have the difference between two perfect squares. This can be rewritten as:

(a^2 + b^2)(a^2 - b^2)

Next, note that the second factor above is ALSO the difference between two perfect squares. Thus, we can rewrite again:

(a^2 + b^2)(a + b)(a - b)

While this is a more complex version of the original, it is worth doing the manipulation this way, as we can now more easily see what type of info we really need.

We can see that Statement 1 plugs into our middle rephrasing, leaving us with:

(a^2 + b^2)(16) as the value we seek. However, as we do not know (a^2 + b^2), we cannot solve. INSUFFICIENT.

Statement 2 gives us PART of the final rephrase we had created above, leaving us with:

(a^2 + b^2)(8)(a - b) as the value we seek. INSUFFICIENT.

It may seem that when we combine, we still cannot solve, because we still have not been able to create (a^2 + b^2). HOWEVER, we actually DO have enough info to determine a and b. This is because:

(a^2 - b^2) = (a + b)(a - b)
16 = 8(a - b)
a - b = 2

As we know that a + b = 8, we know that a = 5 and b = 3. We can thus plug these numbers into the requested form and solve. SUFFICIENT.