Lesson: Exponential Growth

Comment on Exponential Growth

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?
gmat-admin's picture

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

Question link: https://www.beatthegmat.com/0-888-sq-root-0-888-and-0-888-2-in-equality-t274282.html

Hi Brent,
How can a question have 2 correct answer choices? Here A and E both are logically correct.
gmat-admin's picture

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?
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
gmat-admin's picture

In the future please include a link to the question.

Your reasoning is perfectly valid.

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.

I’d gladly add the link for future questions.
But I cooked up this question completely from my doubts.
gmat-admin's picture

Ahhh, good to know!

Cheers,
Brent

Yulia's picture

Hi Brent,

Could you please help me to understand something?

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?
gmat-admin's picture

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

Yulia's picture

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.
gmat-admin's picture

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.

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
gmat-admin's picture

Question link: https://gmatclub.com/forum/is-x-y-1-z-2-y-2-x-z-235918.html

Nice work!
In some cases, you have to rely on a gut feeling about whether a statement is sufficient or not.

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