Tips for Exponents and Roots
Without using a calculator, find the square root of 576. Most people will use a trial-and-error approach, squaring a bunch of numbers until they hit upon the one that works. A savvier approach is to use benchmarks to zero in on the answer:
576 is between 400 and 625, which means that its square root must be between 20 and 25. Since 576 ends in 6, the units digit of its square root must yield a 6 when squared. What digit between 0 and 5 does so? Only 4 does (42 = 16). Therefore, the square root of 576 must be 24. Let's try 289. This lies between 225 and 400, therefore its square root must lie between 15 and 20. What digit between 5 and 10 yields a units digit of 9 when squared? Only 7 does (72 = 49). Therefore, the square root of 289 must be 17.
Of course this method applies only when taking the root of a perfect square (i.e., the square of an integer). But it can also help approximate roots. Let's try 873. This lies between 625 and 900, therefore its root lies between 25 and 30. In fact, it is so close to 900 that its root must be very close to 30. We can estimate its root to be somewhere between 29 and 30. And if we use a calculator, we will find that its root is approximately 29.5.
Another rule to keep in mind is that all even exponents are squares. For example, x2 is obviously a square, but so is x76. How do we know? Because x76 = (x38)(x38). The square root of any even exponent, then, is simply the base raised to half the exponent. For example, , since (x26)(x26) = x52. All we are doing is dividing the exponent by 2. Elaborating on this principle, we can find the fourth root (the square root of the square root) of x52 by dividing the exponent by 4: , because x52 = (x13)(x13)(x13)(x13). Bearing this in mind, if x12 = 2, what is the value of x6? Well, if x12 = 2, then . Therefore, .
Try using these tips in your GMAT preparation. You should find that they save you time and effort, freeing you up for more daunting challenges on the exam.
Tricks Involving the Difference of Two SquaresWhat is the "difference of two squares" anyway? Simply put, it is what it sounds like: one square minus another: . Maybe you have vague recollections of this expression from high-school algebra. If so, perhaps you recall that it can be factored in a predictable pattern: . Basically all we are doing is multiplying the sum of the square roots by the difference of the square roots. It is this pattern that interests the GMAT. On a basic level, the GMAT wants to see that you know how to factor the difference of two squares given in a straightforward form such as . As the exam gets harder, however, you need to be a little more supple in your handling of the difference of two squares.
First, you need to recognize that expressions like can be treated as the difference of two squares, since all even exponents can be considered squares (refer to last week's tip for more details on this): .
Keeping this in mind, we can simplify an expression like by factoring the numerator (top) as the difference of two squares: .
Second, you need to recognize that even expressions like x - y can be factored as the difference of two squares: . Again, all we are doing is multiplying the sum of the square roots by the difference of the square roots. We can use this knowledge to help us simplify expressions like by factoring the numerator as the difference of two squares: .
Third, you need to recognize that you can use the factored form of the difference of two squares to help simplify expressions like (since the GMAT frowns upon radicals in denominators). Since the denominator contains the sum of two radicals, we can multiply top and bottom by the complement () to rid the denominator of the radicals: .
As you prepare for the GMAT, look for opportunities to apply these tricks. If you master them, you will save time and energy on test day.
Tricks with Negative ExponentsFirst, let's review what a negative exponent represents. It is the reciprocal of the corresponding positive power. Let's take, for example, x-1. This can also be expressed as . If we take x-2, we can express it as . The GMAT likes to use negative exponents to hide relationships it wants you to miss.
For example, (x2)(x-2) is not x-4 but rather x0 (remember, when you multiply the same base, you add the exponents), which is equal to 1. How does this come into play on the GMAT? Consider the following question:
What is the value of (x2 + x-2)2 - (x4 + x-4)?
(E) Cannot be determined
Test-takers who are unfamiliar with negative exponents (and algebra generally) will likely choose A or E, figuring either that (x4 + x-4) is the square of (x2 + x-2) or that because there are only variables in the expression, it cannot have an actual value. Both notions are wrong. The answer is C:
(x2 + x-2)2 = (x2 + x-2)(x2 + x-2) = (x2)(x2) + 2(x2)(x-2) + (x-2)(x-2) = x4 + 2(1) + x-4
Therefore, (x2 + x-2)2 - (x4 + x-4) = x4 + x-4 + 2 - (x4 + x-4) = 2.
The GMAT also likes to use negative exponents to test your understanding of the special properties of numbers such as 1 and -1. For example, x = x-1 is the same as x = 1/x, which is the same as x2 = 1. This has two solutions: 1 and -1. Also note that x = x-2 is the same as x = 1/x2, which is the same as x3 = 1, which has only one solution: 1. But x = x2, by contrast, has two solutions: 0 and 1.
Keep your eyes open for negative exponents on the exam. Make sure you know how to manipulate them.
How the GMAT Uses Exponents in Conjunction with Consecutive IntegersFor example, let's say we have an integer x. If we see (x)(x + 1), we know that this is the product of x and the next integer up. But if we want to be slippery, we can express this relationship as x2 + x. All we have done here is distribute the x, but now it is not as clear that we are looking at the product of two consecutive integers.
If we multiply x by the integer just below it, we get (x - 1)(x). To make the multiplication less obvious, we can distribute the x: x2 - x.
If we continue the pattern with (x - 1)(x)(x + 1), we know we have three consecutive integers. But, again, if we want to be slippery, we can distribute: (x2 - x)(x + 1) = x3 + x2 - x2 - x = x3 - x. Now, x3 - x is considerably less recognizable as the product of three consecutive integers.
What use does this have? On the GMAT, these relationships are often used to test your understanding of the concept of odd and even. How? Consider x2 - x, where x is an integer. This expression must be even. First, we can look at it as an odd integer multiplied by an even integer (remember, all pairs of consecutive integers consist of an even number and an odd number), which will always yield an even integer. Or we can look at x itself: if x is even, then x2 - x is even minus even, which is even. If x is odd, then x2 - x is odd minus odd, which is even. Skeptical? Take any two odd numbers and subtract one from the other. The result will always be even.
Now, instead of consecutive integers, let's consider consecutive even or odd integers. If we assume that x is even (or odd), the next even (or odd) integer will be x + 2. We can represent their product as (x)(x + 2) or, less obviously, as x2 + 2x. In contrast to x2 - x, we cannot tell whether x2 + 2x is even or odd. It depends on whether x itself is even or odd. If x is even, x2 + 2x will be even. If x is odd, x2 + 2x will be odd.
These relationships come up much more often in Data Sufficiency than in Problem Solving, for the simple reason that they convey information in a sneaky way. The GMAT uses them as shorthand to see whether you understand the code. Keep these relationships in mind when you practice quantitative problems. There is a very good chance you will see them when you take the exam.
This concludes our series on Roots & Exponents.
Stay tuned for next month's series which takes an in-depth look at the role of the GMAT in the MBA admissions process.
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