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.

If you are looking for extra help in preparing for the GMAT, we offer extensive one-on-one GMAT tutoring for high-achieving students. You can schedule a complimentary, 30-minute consultation call with one of our tutors to learn more! 

Contributor: Elijah Mize (Apex GMAT Instructor)

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Negative Exponents and Negative Bases_
Posted on
30
Aug 2022

Negative Exponents and Negative Bases

Welcome back to our series on exponents. Today we will see what happens when we throw negatives into our exponential expressions. We will explore both negative bases and negative exponents.

First, the bases. The rule to remember for negative bases is that odd powers of negative bases are negative and even powers of negative bases are positive. This rule makes sense when you remember that exponents simply notate a number of multiplications by the base (and remember your rules about multiplication with negative factors). 

Multiplication with an odd number of negative factors yields a negative product: 

(-x)3 = -x * -x * -x

(-x)3 < 0

Multiplication with an even number of negative factors yields a positive product:

(-x)4 = -x * -x * -x * -x

(-x)4 > 0

A note on notation: parentheses should always be used around a negative value as the base of an exponent. If they are not, then the order of operations dictates that the exponent be applied before the negative sign. To avoid confusion, whenever the negative is meant to be left out of the exponential operation, parentheses are used like this: -(x)4 to make the order clear. Please note that -(x)4 is less than 0, and (-x)4 is greater than 0.

Now for negative exponents. Here is a simple rule is the best way to explain it:

x-n = 1/xn

A negative exponent indicates a value reciprocal to the value with a positive exponent. It’s good practice to “translate” any exponential expressions with negative exponents to their reciprocal positive forms. Seeing it both ways can help you make sense of problems.

5-4 = 1 / (54) = 1 / 625

17-2 = 1 / (172) = 1 / 289

(9 / 16)-2 = (16 / 9)2 = 256 / 81

Integer bases with negative exponents go under a numerator of 1; fractional bases with negative exponents simply flip. Let’s look at some “double negative” exponential expressions.

(-6)-3 = 1 / (-6)3 = 1 / -216

(-2)-10 = 1 / (210) = 1 / 1024

(-4 / 3)-4 = (-¾)4 = 81 / 256

Let’s get into some official GMAT problems. Be careful with this first one!

From the consecutive integers -10 to 10 inclusive, 20 integers are randomly chosen with repetitions allowed. What is the least possible value of the product of the 20 integers?

  1. (-10)20
  2. (-10)10
  3. 0
  4. -(10)19
  5. -(10)20

One incorrect answer is chosen far more often than any other on this problem: answer choice D. Trying to minimize the product, many people consider taking the maximum number (20) of the lowest value (-10). The common mistake is then thinking that -(10)20 (answer choice E) is actually a positive value since it involves an even number (20) of negative factors. Many people then take “the next best thing” in answer choice D, which shifts the exponent to the next odd number down from 20.

In fact, -(10)20 does not involve an even number of negative factors, since the negative sign is excluded from the exponential expression by the parentheses. Answer E means “take 1020 and make it negative.” It is true that taking 20 negative tens and multiplying them all together produces a large positive value (the opposite of what we are aiming for on this problem), but this misguided idea is notated by answer choice A – not answer choice E. Remember that the answer choices notate the product of the 20 factors, not necessarily a condensed list of the 20 factors. 

It is possible to choose 20 integers from -10 to 10 inclusive that, when multiplied, yield a product of -(10)20 (answer choice E). The “least possible value” is obtained by finding the greatest absolute value (distance from 0) in negative form. So we want all 20 of our factors to be either 10 or -10 since this will maximize the absolute value (distance from 0) of the product. To ensure that the product is also negative, we simply need an odd number of negative tens. We can use nineteen negative 10s and 1 positive 10, 1 negative 10 and 19 positive 10s, or any odd combination in between. Any of these options will yield a product of -(10)20. Read the notation carefully!

Let’s try another:

The value of 2(-14)+2(-15)+2(-16)+2(-17)/5 is how many time the value of 2(-17)?

  1. 3/2
  2. 5/2
  3. 3
  4. 4
  5. 5

This problem benefits from the skill of noticing patterns and “checking” them. You should see the pattern in the numerator and generalize it by saying “the negative exponent on the 2 keeps decreasing by 1.” Then you can see how this pattern “works” by checking a single case. 

2-17 = 1 / 217

2-16 = 1 / 216

Since 217 = 2 * 216, (1 / 217) is half the value of (1 / 216). Or, to say it a more useful way, 2-16 = 2 * 2-17 This pattern will continue through the numerator. Since we are looking for how many copies of 2-17 we have in this expression, we can replace 2-17 with 1 and follow the pattern.

(214 + 2-15 + 2-16 + 2-17) / 3

(8 + 4 + 2 + 1) / 3

15 / 3 = 3

And the correct answer is C.

Here’s a final problem for today:

a is a nonzero integer. Is 

a2greater than 1?

  • a < -1
  • a is even.

To evaluate statement 1, simply start by checking a = -2. (-2)-2 = 1 / (-2)2  = ¼. Moving on to a = -3, (-3)-3 = 1 / (-3)3 = 1 / -27. This time the value is negative, but the positive value from a = -2 is still less than 1. If you imagine continuing with a = -4, a = -5, etc., you will just keep making smaller and smaller fractions. Statement 1 alone is sufficient.

Statement 2 on its own is easy to check since we already know from checking statement 1 that some even values for the variable a yield an aa with a value less than 1. And it shouldn’t be hard to imagine an even value for variable a where aa is greater than 1. For example, 22 = 4 and 44 = 256. So statement 2 alone is insufficient, and the correct answer is A.

If you went with answer choice C, here’s what might have happened. Noticing that statement 1 tells you the base is negative, you might have seen next that statement 2 tells you the exponent is even. You might have thought that this “evenness” of the exponent makes the difference since it determines the positivity or negativity of the expression. Very often in DS problems with negative bases, the even/odd identity of the exponent really matters. But in this case, it’s a trap, because we were asked whether aa is greater than 1 (not 0), and the fact that the exponent is also negative means that it’s even/odd identity is irrelevant – the value is always less than 1.

The rules governing negative exponents and negative bases are simple, but the GMAT and EA problems that employ these rules can catch you if you aren’t careful. Next time we will look at another tricky exponential scenario: when the base is between -1 and 1.

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Contributor: Elijah Mize (Apex GMAT Instructor)

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