Understanding Square Roots and Differences in Mathematics

Which expression is NOT equivalent to the square root of the difference between two numbers a and b?

• A. (√a) - (√b)
• B. √(a - b)
• C. √a - √b
• D. √(ab)

Answer: (B)

The square root of the difference between two numbers, denoted as √(a - b), represents the square root of the value obtained when the smaller number, b, is subtracted from the larger number, a. To determine which expression is not equivalent to this, it's essential to consider the properties of square roots and differences. The square root function does not distribute over subtraction. Therefore, the expression (√a) - (√b) and its equivalent formulation, √a - √b, suggest that both represent the individual square roots of a and b subtracted from each other, not the square root of their difference. Additionally, √(ab) represents the square root of the product of a and b, which also does not correspond to the square root of the difference between the two numbers. Focusing on option B, √(a - b) is precisely the expression given in the question. It is not just another representation but is, in fact, equal to itself. Thus, it is not equivalent to any of the other expressions provided, particularly those that manipulate the square roots individually rather than the difference of a and b as a singular entity. In summary, since √(a - b) represents the square

When it comes to the Graduate Management Admission Test (GMAT), mastering the nuances of mathematical concepts can make a world of difference in your preparation. One such concept is the square root of the difference between two numbers. Let’s break it down, keeping it clear and engaging!

What's the Problem?

Imagine you’re faced with a question like this: Which expression is NOT equivalent to the square root of the difference between two numbers ( a ) and ( b )? You’ve got some options. It’s a bit tricky, right? Let’s look at them:

A. ( (\sqrt{a}) - (\sqrt{b}) )

B. ( \sqrt{(a - b)} )

C. ( \sqrt{a} - \sqrt{b} )

D. ( \sqrt{(ab)} )

The correct answer? It’s ( \sqrt{(a - b)} ). But why?

Dissecting the Square Roots

First off, let’s clarify what ( \sqrt{(a - b)} ) actually represents. In simple terms, it’s the square root of whatever number you get when you subtract ( b ) from ( a ). If ( a ) is bigger than ( b ), fantastic! You've got a positive result. But if ( b ) is greater than ( a ), well, we’ve wandered into imaginary number territory! Yet, that’s not our focus right now.

Now, on to the other expressions. Here’s the kicker: square roots and subtraction can be a bit of a danger zone if you don’t play by the math rules. See, the square root function doesn’t just distribute over subtraction like chocolate chips in a cookie dough.

For instance, both ( (\sqrt{a}) - (\sqrt{b}) ) and its lookalike ( \sqrt{a} - \sqrt{b} ) suggest that you’re subtracting the individual square roots—which isn’t the same as taking the square root of the difference. They’re two entirely different mathematical animals! So, they can't be equivalents here.

Let’s Not Forget the Product!

Then there’s option D: ( \sqrt{(ab)} ). Yep, that’s the square root of the product of ( a ) and ( b ). Great for some problems but completely irrelevant for this particular question. It doesn’t matter how much you multiply; it won’t yield the square root of the difference.

Digging deeper, focusing on option B, ( \sqrt{(a - b)} ) essentially rings true to itself. It’s not just another expression; it's the original one we’re discussing. If anything, it doesn’t show up as an equivalent to the others, especially those that are each dancing around the individual square roots rather than the difference.

Wrapping It Up

In a nutshell, understanding the properties of square roots and differences can help illuminate your reasoning when tackling similar problems on the GMAT. Having a strong grasp enables you to not only recognize the correct answers but also understand why they’re correct.

So, as you prepare for your GMAT, keep your mind sharp and remember: Math isn’t just about numbers; it’s about interpreting what they represent. Keep practicing and approaching these concepts with curiosity, and you’ll find yourself well-equipped for the challenge ahead. You’ve got this!