Roots are Exponents

October 7, 2010

When you first approach GMAT math, the number of topics is bewildering. For example, Total GMAT Math has chapters on over 40 different content areas.

All that math starts to get easier when you recognize the relationships between content areas. I've written before about the links between rates, ratios, and percents. Despite the different names, they are really the same.

Another great example of this phenomenon is the link between roots (most often, square roots) and exponents.

A root is essentially an exponent in reverse. For example, the square root of 2 (I'll write it here as "rt(2)") is the same as 2^1/2. Don't always convert one way or the other: Sometimes you'll want the root, sometimes the exponent.

Similarly, the third root of x--³rt(x)--is equivalent to x^1/3. Third roots are very difficult to calculate by hand, so any time you see one on the GMAT, it's almost certain that you'll be asked either to approximate or to simplify instead.

Here's an example. Let's say you're asked for the third root of 4⁶. No one in their right mind will try to calculate 4⁶, let alone take the third root.

Given that the third root is the same as the one-third power, we know that we're looking for (4⁶)^1/3. Now we can apply a familiar exponent rule to combine the exponents. One exponent raised to another means that we should multiply them.

6 times 1/3 is 2, so the expression simplifies to 4², or 16. Not so bad in the end, eh?

Any time you see exponents and roots in the same GMAT question, don't fret--remember that it's really only one concept. Perhaps the exponents and roots can work together to make the problem that much easier to solve.

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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