For all a and b where (a^2 - b^2)(a - b) is defined, the expression is

STRATEGY: Upon reading any GMAT Problem Solving question, we should always ask, Can I use the answer choices to my advantage?
In this case, we can test for equivalency.
Now let's give ourselves up to 20 seconds to identify a faster approach.
In this case, we can also try to manipulate the original expression into the form of one of the answers choices.
On test day I'd typically test for equivalency, but let's examine both approaches

APPROACH #1: Testing for equivalency
Key concept: If two expressions are equivalent, they must evaluate to the same value for every possible value of x.
For example, since the expression 2x + 3x is equivalent to the expression 5x, the two expressions will evaluate to the same number for every value of x.
So, if x = 7, the expression 2x + 3x = 2(7) + 3(7) = 14 + 21 = 35, and the expression 5x = 5(7) = 35

More strategy: The expression contains \(\sqrt{a^2 - b^2}\), closely resembles what we get when applying the Pythagorean theorem.
So I'm going to test \(a = 5\) and \(b = 4\), since they are two values of the Pythagorean triple 3-4-5

When \(a = 5\) and \(b = 4\), we can calculate the value of the given expression as follows: \((\sqrt{a^2 - b^2})(\sqrt{a - b}) = (\sqrt{5^2 - 4^2})(\sqrt{5 - 4}) = (\sqrt{9})(\sqrt{1}) = (3)(1) = 3\)

So, the correct answer will also evaluate to 3, when \(a = 5\) and \(b = 4\)

Plug \(a = 5\) and \(b = 4\) into the five answer choices....

A. \((5 - 4)(\sqrt{5 - 4}) = 1\). We need 3. Eliminate.

B. \((5 - 4)(\sqrt{5 + 4}) = 3\). Perfect! Keep.

C. \((5 - 4)^2 = 1\). We need 3. Eliminate.

D. \(5^3 - 4^3 = 61\). We need 3. Eliminate.

E. \(\sqrt{5^3 - 4^3} = \sqrt{61}\). We need 3. Eliminate.

By the process of elimination, the correct answer is B


APPROACH #2: Apply some algebra

Given: \((\sqrt{a^2 - b^2})(\sqrt{a - b})\)

Factor the first part: \((\sqrt{(a + b)(a - b)})(\sqrt{a - b})\)

Useful property: \((\sqrt{x})(\sqrt{y}) = \sqrt{xy}\)

When we apply the above property we get: \(\sqrt{(a + b)(a - b)(a - b)}\)

Rewrite as follows: \(\sqrt{(a - b)^2(a + b)}\)

Now rewrite as: \((\sqrt{(a - b)^2})(\sqrt{(a + b)})\)

Simplify: \((a - b)\sqrt{(a + b)})\)

Answer: B