What is the value of (3/2)^(1/2) - (2/3)^(1/2) ?

Exponents/radicals are one of those math concepts that you ABSOLUTELY should make sure you know the rules. However, we can also get a lot of them right just by ballparking. Let's do this question both ways.

First, following the rules:
Original: \(\sqrt{\frac{3}{2}} - \sqrt{\frac{2}{3}}\)
Convert to \(\frac{\sqrt{3}}{\sqrt{2}}-\frac{\sqrt{2}}{\sqrt{3}}\)
Multiply first fraction by \(\frac{\sqrt{2}}{\sqrt{2}}\) and second fraction by \(\frac{\sqrt{3}}{\sqrt{3}}\): \(\frac{\sqrt{3}\sqrt{2}}{2}-\frac{\sqrt{3}\sqrt{2}}{3}\)
Multiply first fraction by \(\frac{3}{3}\) and second fraction by \(\frac{2}{2}\) to get common denominator: \(\frac{3\sqrt{6}}{6}-\frac{2\sqrt{6}}{6}\)
Subtract second fraction from first: \(\frac{\sqrt{6}}{6}\)
That's what I'd have expected to see in the answer choices, but it's not there. If we multiply that by \(\frac{\sqrt{6}}{\sqrt{6}}\): \(\frac{6}{6\sqrt{6}}\) which is \(1/\sqrt{6}\)
Answer choice B.

And now for ballparking:
\(\sqrt{2}\) is roughly 1.4. \(\sqrt{3}\) is roughly 1.7. Those are worth memorizing because they are useful a LOT in ballparking, especially on geometry and exponents questions. Plugging those into the question stem, we have:
\(\frac{1.7}{1.4}-\frac{1.4}{1.7}\)
Let's ballpark that to be 1.2 - 0.8.
That's 0.4
Now, check the answer choices (for \(\sqrt{6}\), we know that is greater than \(\sqrt{4}\) and less than \(\sqrt{9}\), so it is between 2 and 3; let's go ahead and use 2.5).
(A) (1.7-1.4)/2.5 = 0.3/2.5 Way too small. Get rid of it.
(B) 1/2.5 = 0.4 Keep it.
(C) 1.7/3 Greater than 1/2. Get rid of it.
(D) 1.7/2 Way too big. Get rid of it.
(E) Something even greater than 1.7/2 Way too big. Get rid of it.
Answer choice B.

Either way works. I tend to think you're less likely to make a mistake using the second method and it's at least as fast, but if you're super comfortable with exponents and radicals, have a good time with the first method. Then again, if you're super comfortable with exponents and radicals, you shouldn't need an explanation for this question, anyway!


ThatDudeKnowsBallparking
ThatDudeKnowsExponentsRadicals