Posted on
08
Sep 2022

## Exponent Properties in Data Sufficiency

Exponents have properties that make them ideal pieces for data sufficiency problems on the GMAT and Executive Assessment (EA) quant sections. We’ve surveyed most of these properties in our first four articles in this series, but a couple of key ones haven’t been mentioned explicitly.

First, x0 = 1. A full mathematical explanation of this property is beyond the scope of this series of articles and is unnecessary for GMAT/EA preparation. If you know the rule, you can employ it as needed. The debated exception 00 exists but does not occur on GMAT quant.

Second, the numbers 0 and 1 are both “immune” to exponents. We mentioned this rule for 1 in our last article, but it is important to know that the rule applies to 0 as well. 0x = 0, and 1x = 1.

Another important rule for exponent DS problems has already been mentioned: even powers of negative numbers are positive, and odd powers of negative numbers are odd. To extend this, xeven is always greater than or equal to 0 (it is only equal to 0 when x = 0; in all other cases xeven is positive).

A basic rule to remember is that positive numbers have two square roots which are negative and positive versions of the same value. If b2 = 16, b may equal 4 or -4. Forgetting about the possibility of the negative square root usually leads to incorrect answers.

One more important rule to remember is that if x is positive, then xy is positive. No exponent can cause an exponential expression with a positive base to have a negative value or a value of 0. To state the same rule differently, exponential expressions with negative values have negative bases.

Let’s get into some official problems:

Is zp negative?

1. pz4 < 0
2. p + z4 = 14

We will need to know the signs of the variables p and z. If they are both positive or both negative, then the product zp is positive. If z or p is negative and the other positive, then the product zp is negative.

Statement 1 tells us that the product p * z4 is negative. Therefore either of p or z4 is negative and the other is positive. Since the exponent 4 is positive, z4 is always positive (or 0, but the statement rules out that possibility). This means that p must be negative. This isn’t enough to answer whether zp is negative, but it may be useful if we have to combine statements 1 and 2.

Statement 2 tells us that the sum of p and z4 equals 14. There are simply too many possibilities for this statement on its own to be sufficient.

Now we must combine statements 1 and 2. We know from statement 1 that p is negative. Therefore z4 must be positive in order for p + z4 to have the positive sum of 14. But we already know that z4 is positive because the exponent 4 is even! A common mistake in these DS problems is to accidentally transfer information about z4 back to the variable z itself. Doing so on this problem might lead you to incorrect answer choice C. In this case, statements 1 and 2 together are still not sufficient, and the correct answer is E.

Let’s try another:

If r, s and t are nonzero integers is r5s3t4 negative?

• rt is negative.
• s is negative.

Here’s another even exponent of 4. This means that t4 is either positive, or 0 in the case of t = 0. If t = 0, then the product r5s3t4 is 0 and not negative.

Statement 1 tells us that the product rt is negative. Therefore either of r or t is negative and the other is positive. We still know nothing about the variable s, so this can’t be sufficient on its own.

Statement 2 tells us that s, and therefore s3, is negative, but it says nothing about r or t. This can’t be sufficient on its own.

Combining the statements, we can approach statement 1 with two pieces of info: that t4 is positive (or 0) and that s3 is negative (since s is negative). Therefore the product r5s3t4 is (?) * (-) * (+), and the sign of the variable r makes all the difference.

Returning to statement 1, we consider again that the product rt is negative. This means that either r or t is negative, but unfortunately, we don’t know which one. Again, if we mistakenly jump from the fact “t4 is positive” to the unfounded conclusion “t is positive,” we will select incorrect answer choice C. Again, the statements together are insufficient, and the correct answer is E.

Here’s another:

If a and b are integers is a5< 4b?

1. a3= -27
2. b2= 16

The upshot of statement 1 is that a is negative. Therefore a5 is also negative, since 5 is another negative exponent. To get specific (which is probably unnecessary for the problem), a = -3 and a5 = -243.

We don’t know anything about b, so it’s tempting to conclude that statement 1 alone is insufficient. But we are asked to compare a5 against 4b, not against b itself. If you remember the rule that exponential expressions with positive bases have positive values, you’ll see that statement 1 alone is sufficient. a5 is negative, and 4b is positive.

Statement 2 tells us that b = 4 or -4. Therefore 4b = 256 or 1/256. Either way, without statement 1, we know nothing about a5. Statement 2 on its own is insufficient, and the correct answer is A.

Here’s a final DS exponents problem:

If x and y are integers, is x > y?

1. x + y > 0
2. yx < 0

Statement 1 is certainly insufficient by itself, but we should still think about what it tells us. Either x and y are both positive, or one is negative and the other is positive, with the positive number having the greater absolute value.

Statement 2 tells us that yx is negative. The rule we need is that exponential expressions with negative values have negative bases. This means y is negative.  On its own, this statement is still insufficient. But combined with statement 1, we know that x must be greater than y, because x must be positive in order to produce a positive sum with a negative number y. The statements together are sufficient, and the correct answer is C.

We are halfway through our series on exponents and have covered all the basics. The remaining five articles will cover specific problem types involving exponents.

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Contributor: Elijah Mize (Apex GMAT Instructor)

Posted on
07
Sep 2022

## Bases Between -1 and 1

Many GMAT and Executive Assessment (EA) exponent problems – especially data sufficiency ones – require you to consider fractional bases. By this I mean proper fractions with values between -1 and 1, not improper fractions whose numerators exceed their denominators.

There are four “kinds” of bases separated by the three “boundary points” of -1, 0, and 1. Numbers in each of the four “zones” separated by these values behave similarly as bases of exponents. On many DS problems, we need to consider numbers less than -1, negative fractions, positive fractions, and numbers greater than 1, as well as the boundary points of -1, 0, and 1. This sounds like a lot of work, but practice will build your “spidey sense” for when a certain kind of base leads to an exception and an insufficient statement.

Let’s handle the “boundary points” first. As mentioned in a prior article, the boundary point of 1 is simple because it is “immune” to exponents: 1anything = 1. Likewise, 0positive = 0. For -1, (-1)even = 1 and (-1)odd = -1. Since (-1)anything equals either 1 or -1, it is worth noting that the absolute value of -1anything is 1. If you are unfamiliar with absolute value, don’t worry, it’s a simple concept: absolute value is a number’s distance from 0 on a number line. The absolute value of a positive number is . . . itself. The absolute value of a negative number is simply the positive version of the number. For positive numbers, “normal” value and absolute value are the same and trend together. For negative numbers, absolute value increases as “normal” value decreases. Absolute value is notated with vertical lines on either side of a value, variable, or expression. |(-1)anything| = 1.

Now for the four ranges of numbers. Numbers greater than 1 are the simplest. For these numbers, the higher the exponent, the higher the overall value. For numbers greater than 1, higher powers have higher values.  Numbers less than -1 are only slightly more complex. For numbers less than -1, higher powers have higher absolute values, but odd powers are negative and even powers are positive.

Now for the positive and negative fractions: the more times you multiply a fraction by itself, the closer the resulting value gets to 0. (¾)2 or ¾ * ¾, which we can read as “¾ of ¾,” is less than ¾. For positive fractions, higher powers have lower values. For negative fractions, higher powers have lower absolute values, but odd powers are negative and even powers are positive.

Let’s demonstrate all of our rules with the examples of 2, -2, ½, and -½.

2 < 22 < 23 . . .

|2| < |22| < |23| . . .

(-2)5 < (-2)3 < (-2) < 0 < 22 < 24 < 26 . . .

|-2| < |(-2)2| < |(-2)3| . . .

½ > (½)2 > (½)3 . . .

|½| > |(½)2| > |(½)3| . . .

-½ < (-½)3 < (-½)5 < 0 < (-½)6 < (-½)4 < (-½)2

|-½| > |(-½)2| > |(-½)3| . . .

The patterns for the negatives can take some getting used to, so study these rules frequently and, more importantly, build your fluency with practice problems! Here’s a straightforward one:

Is x2 greater than?

1. x2 is greater than 1.
2. x is greater than -1.

For statement 1, it helps to remember that only numbers greater than 1 or less than -1 can have powers greater than 1. Powers of fractions are always fractions. So if x2 is greater than 1, x is either greater than 1 or less than -1. If x is less than -1, then x2, which according to the statement is greater than 1, is greater than x. And if x is greater than 1, it still gets larger when it is squared, so x2 is always greater than x, and statement 1 is sufficient.

Statement 2 tells us that x is greater than -1. If we remember our boundary points, we can solve this one without having to think about fractions. X could be 0 or 1, and in either case, x2 is equal to, not greater than, x. But for any number greater than 1, x2 is greater than x. So statement 2 on its own is insufficient, and the correct answer is A.

Let’s try another DS problem:

Is xy > x2y2?

1. 0 < x2 < 1/4
2. 0 <  y2 < 1/9

To verbalize the question: is the absolute value of the product xy greater than the square of the product xy?

Statement 1 tells us that x2 is a positive fraction, which means that x itself is a fraction with a greater absolute value, but we don’t know whether it is positive or negative. Without knowing anything about y, this isn’t enough. Statement 1 alone is insufficient. Statement 2 is similar and also insufficient. Taking the statements together, we know that the absolute value of x is greater than x2, and the absolute value of y is greater than y2. It follows that the absolute value of the product xy is greater than x2y2. We don’t know whether x and y are positive or negative, but we’re talking absolute value so it doesn’t matter. The statements together are sufficient, and the correct answer is C.

Finally, let’s see what happens when fractional bases meet negative exponents:

(1/2)-3(1/4)-2(1/16)-1=

A. (1/2)(-48)

B. (1/2)(-11)

C. (1/2)(-6)

D. (1/8)(-11)

E. (1/8)(-6)

This problem benefits from “translating” out of the negative-exponent form. (½)-3 = 23, (¼)-2 = 42, and (1/16)-1 = 16. Ideally, you recognize that everything in the expression can be converted to powers of 2. 42 and 16 both equal 24, so the full expression is 23 * 24 * 24, or, remembering your exponent rules, 211. If you noticed how the fractional base and the negative exponent “canceled” each other, you should recognize 211 as (½)-11, answer choice B.

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Contributor: Elijah Mize (Apex GMAT Instructor)