# Lesson: Exponential Growth

## Comment on Exponential Growth

### Hi - regarding the below

Hi - regarding the below question from
GMAT practice question (difficulty level: 500 to 650) - Math Revolution

Which of the following is closest to (-3/4)^199？
A. -1
B. -1/2
C. 0
D. 1
E. 2

You guys say the OA is C... I understand that as the fraction's exponent increases, the magnitude will decrease and continue approaching zero. But given that the exponent is odd shouldn't the answer be B? ### You're right in that the odd

You're right in that the odd exponent means (-3/4)^199 equals some negative number. However that negative number is very close to 0.

Using a scientific calculator (which isn't provided on test day) we can calculate this.

We get: (-3/4)^199 ≈ -0.000000000000000000000000137

As you can see, this number if a lot closer to 0 than it is to -1/2

Hi Brent,
How can a question have 2 correct answer choices? Here A and E both are logically correct. ### That question on BTG is

That question on BTG is poorly transcribed. I have changed the link to go here: https://gmatclub.com/forum/if-x-0-888-y-0-888-1-2-and-z-0-888-2-then-whi...

In the GMAT Club post, the answer choices are correctly transcribed.

Cheers,
Brent

### Ques.) X*Y = positive number?

Ques.) X*Y = positive number?
Statement 1.) X^2 > 0 → it means a.) Anything greater than 0 is +ive value. b.) it could be possible that X = -ive value but becomes +ive value when squared. Therefore, Insufficient.

Statement 2.) |X| > 0 → means X = +ive value when inside |Modulus| [Like when X = +ive value when squared] . X without Modulus could be a -ive value. Since X exists in question’s equation without Modulus. Therefore, we can’t say X = +ive with certainty.

If X^2 or |X| is given we cannot say it with certainty whether their value is Positive or Negative.
Pls correct me If I’m wrong anywhere ### In the future please include

If we know that x² > 0 and |x| > 0, there's no way to determine whether x is positive or negative.
For example, if x = 1, then it's true that x² > 0 and |x| > 0.
Likewise, if x = -1, then it's also true that x² > 0 and |x| > 0.

Aside: If the target question asks "Is xy positive?", then we can quickly see that the statements combined are not sufficient, since we aren't provided any information about y.

But I cooked up this question completely from my doubts. ### Ahhh, good to know!

Ahhh, good to know!

Cheers,
Brent ### Hi Brent,

Hi Brent,

I dont understand the written rule about Negative bases x<-1.

I understand as follows:
If exponent is even then the value of X^n increases as even exponent increases

If exponent is odd then the value of X^n decreases as add exponent increases.

Is my understanding correct? ### There are four different

There are four different cases (not two case)
These four cases are summarized at the end of the above video. They are as follows:

THE BASE (x) IS POSITIVE
Case i: The base is between 0 and 1
The value of x^n gets closer and closer to zero as n increases.
For example:
(0.1)^1 = 0.1
(0.1)^2 = 0.01
(0.1)^3 = 0.001
(0.1)^4 = 0.0001
etc

Case ii: The base is greater than 1
The value of x^n increases as n increases.
For example:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
etc

------------------------

THE BASE (x) IS NEGATIVE
Case iii: The base is between -1 and 0
The value of x^n OSCILLATES between positive and negative, and the MAGNITUDE of x^n gets closer and closer to zero as n increases.
For example:
(-0.1)^1 = -0.1
(-0.1)^2 = 0.01
(-0.1)^3 = -0.001
(-0.1)^4 = 0.0001
(-0.1)^5 = -0.00001
(-0.1)^6 = 0.000001
etc

Case iv: The base is less than -1
The value of x^n OSCILLATES between positive and negative, and the MAGNITUDE of x^n gets bigger as n increases.
For example:
(-2)^1 = -2
(-2)^2 = 4
(-2)^3 = -8
(-2)^4 = 16
(-2)^5 = -32
(-2)^6 = 64
etc ### Hi Brent,

Hi Brent,

Thank you for the reply. I have confusion with the negative base only.
Case iii: The base is between -1 and 0. The value of x^n will get closer to zero regardless if the exponent is even or odd?

Case iv: The base is less than -1. The value of x^n will get bigger if the exponent is even but if the exponent is odd the value is getting smaller. Is it correct? Ex. (-2)^1 = -2 and (-2)^3 = -8 in which -2 > -8, so I concluded that when N is odd the magnitude of x^n is getting smaller. ### Case iii: The base is between

Case iii: The base is between -1 and 0
You're right; I should have said: if the base is between -1 and 0, then the value of x^n OSCILLATES between positive and negative, and x^n gets closer to zero as the value of n increases.

Case iv: The base is less than -1
Magnitude refers to a number's DISTANCE from zero on the number line.
So, the MAGNITUDE of -10 is greater than the MAGNITUDE of 3.
So, the value of x^n OSCILLATES between positive and negative, and the MAGNITUDE of x^n gets bigger as n increases.

### Hi Brent, I'm a bit confused

Hi Brent, I'm a bit confused with negative base in exponent. As the MAGNITUDE of x^n gets closer and closer to zero as n increases as below. Does it mean the number with more zero in the front is bigger than number with less zero in the front? e.g -0.001 > -0.1 ? same with positive 0.001 > 0.1 ?

For example:
(-0.1)^1 = -0.1
(-0.1)^2 = 0.01
(-0.1)^3 = -0.001
(-0.1)^4 = 0.0001
(-0.1)^5 = -0.00001
(-0.1)^6 = 0.000001 ### With two POSITIVE numbers,

With two POSITIVE numbers, the value FARTHEST from zero on the number line will be the largest number.
For example: 4 < 5, because 5 is further from zero than 4 is.

With two NEGATIVE numbers, the value CLOSEST to zero on the number line will be the largest number.
For example: -6 < -1, because -1 is closer to zero than -6 is.

The same thing applies to decimal values.
For example, 0.0001 < 0.1, because 0.1 is further from zero than 0.0001 is.
Conversely, we know that -0.001 > -0.1, because -0.001 is closer to zero than -0.1 is.

### Thanks Brent. Great

Thanks Brent. Great explanation. Now this makes sense that -0.027 > -0.3

### that 650-800 I got it right,

that 650-800 I got it right, but I just doubt it insufficient cuz z could be negative OR positive, we don't know what tends out to be true so E 