Data Sufficiency

LulaBrazilia wrote:What is the value of a� - b�?

1) a² - b² = 16
2) a + b = 8

Target question: What is the value of a� - b�?

NOTE: a� - b� is a difference of square, which we can factor. a� - b� = (a² - b²)(a² + b²)
So, we can REPHRASE the target question as...

REPHRASED target question: What is the value of (a² - b²)(a² + b²)?

Statement 1: a² - b² = 16
Okay, so we know the value of HALF of the target expression to get (16)(a² + b²), but we still don't know the value of a² + b², so statement 1 is NOT SUFFICIENT

We can also demonstrate that statement 1 is NOT SUFFICIENT by finding values of a and b that satisfy this condition. Here are two:
Case a: a = 4 and b = 0, in which case (a² - b²)(a² + b²) = (16)(16) = 256
Case b: a = √17 and b = 1, in which case (a² - b²)(a² + b²) = (16)(18) = something other than 256
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: a + b = 8
There are several values of a and b that satisfy this condition. Here are two:
Case a: a = 8 and b = 0, in which case (a² - b²)(a² + b²) = (64)(64) = 64²
Case b: a = 4 and b = 4, in which case (a² - b²)(a² + b²) = (0)(32) = 0
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 says that a² - b² = 16
Since we can factor a² - b² to get (a + b)(a - b), statement 1 is actually telling us that (a + b)(a - b) = 16
Statement 2 says that a + b = 8
So, let's take (a + b)(a - b) = 16 and replace (a+b) with 8 to get: (8)(a - b) = 16
This means that (a - b) = 2

At this point, we're done.
We now know that a - b = 2 AND we know that a + b = 8
Here we have 2 linear equations, which we COULD solve for a and b.
Once we know the exact values of a and b, we can definitely determine the value of (a² - b²)(a² + b²)
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

Answer = C


Aside: If we solve the system, a - b = 2 and a + b = 8, we get a = 5 and b = 3. This means that (5² - 3²)(5² + 3²) = (16)(34)


Cheers,
Brent

Hi LulaBrazilia,

The other posts in this thread have pointed out that this question is based on Classic Quadratics. As you do better and better in the Quant or Verbal sections, the GMAT is more likely to test you on patterns that you probably KNOW but in a format that you're not used to seeing.

Most people know that X^2 - Y^2 = (X+Y)(X-Y)

When looking at A^4 - B^4, you should be reminded of the above Quadratic Formula...

A^4 - B^4 = (A^2 + B^2)(A^2 - B^2)

From here, the pattern (and what you could do next) is a bit easier to recognize.

As a takeaway, when you're dealing with something that "looks strange", try thinking about what it reminds you of and what category of Quant or Verbal it's based on.

GMAT assassins aren't born, they're made,
Rich