Lesson: Negative Exponents

Comment on Negative Exponents

Here in the 3rd question, how do we know it is 3^(x+1). I actually interpreted it as (3^x)+1. Don't you think it is a bit confusing?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/x-is-a-negative-integer-1895.html

The GRE Prep Club website has a few minor formatting issues when it comes to exponents. I agree, it's not 100% clear whether it's 3^(x+1) or 3^x + 1.

On test day, the notation will not be so ambiguous.

Cheers,
Brent

What allows us to “flip each fraction “ in your process for the 3rd to last problem link?
greenlight-admin's picture

Hi Jay,

Are you referring to a question in the above video, or are you referring to one of the linked practice questions in the Reinforcement Activities box?

In general, we divide by a fraction by multiplying by its reciprocal.

For example, 1/2 ÷ 3/5 = 1/2 x 5/3
Also, 2/7 ÷ 11/5 = 2/7 x 5/11

Likewise, (2/3)/(7/11) = 2/3 ÷ 7/11 = 2/3 x 11/7

Does that help?

Cheers,
Brent

Specifically I’m speaking of the 3rd to last reinforcement problem. I get multiplying by the reciprical for division of fractions, but I didn’t see the need for that in this problem, however when you described the process you said flip the equation and I was wondering how that was possible. Thanks as always

Jason
greenlight-admin's picture

Question link: https://greprepclub.com/forum/if-5n-4n-x-8511.html

We have the equation 5n/(4n - x) = 1/0.788

I wanted to apply the rule that says (a - b)/c = a/c - b/c
However, to apply this law, I needed to create an equivalent equation featuring a SINGLE expression in the denominator. That is, I wanted to rewrite 5n/(4n - x) as (4n - x)/5n

To accomplish this, I used the fact that, if two fractions are equal, then their reciprocals must also be equal.

That is, if x/y = a/b, then it must also be true that y/x = b/a (as long as none of the values is zero)

Does that help?

Cheers,
Brent

https://greprepclub.com/forum/if-5n-4n-x-8511.html

In this input type question, can I enter my answer in decimal form (0.06) if nothing is mentioned?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/if-5n-4n-x-8511.html

Great question!

On the GRE, you will be told when to enter your response as a fraction. You will be given two boxes (one box for the numerator and one box for the denominator) to enter values. If there's just one box, then you can enter the value as a decimal.

When I first posted the question, I forgot to add directions to submit the answer as a fraction. I have now added text to that effect.

For more on how to enter values in Numeric Entry questions, watch 2:00 to 5:10 of the following video: https://www.greenlighttestprep.com/module/general-gre-info-and-strategie...

Cheers,
Brent

t is an integer

Quantity A: 1/(1 + 2^t)
Quantity B: 1/(1 + 3^t)


We can solve this question using matching operations
Given:
Quantity A: 1/(1 + 2^t)
Quantity B: 1/(1 + 3^t)

Since 2^t is POSITIVE for all integer values of t, we know that 1 + 2^t is also POSITIVE
This means we can safely multiply both quantities by (1 + 2^t) to get:
Quantity A: 1
Quantity B: (1 + 2^t)/(1 + 3^t)

Likewise, since (1 + 3^t) is POSITIVE, we can safely multiply both quantities by (1 + 3^t) to get:
Quantity A: 1 + 3^t
Quantity B: 1 + 2^t

Subtract 1 from both quantities to get:
Quantity A: 3^t
Quantity B: 2^t

From here, we can TEST some integer values of t

If t = 0, we get:
Quantity A: 3^0 = 1
Quantity B: 2^0 = 1
In this case, the two quantities are EQUAL

If t = 1, we get:
Quantity A: 3^1 = 3
Quantity B: 2^1 = 2
In this case, Quantity A is GREATER

Answer: D
----------------------
In this question how do we know that 1/1+2^t is positive or negative?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/t-is-an-integer-9385.html

Since 2 is positive, we can conclude that 2^t is POSITIVE for all values of t

NOTE: Many students (incorrectly) believe that a NEGATIVE exponent results in a NEGATIVE value. This is not true.

For example, 2^(-1) = 1/2
2^(-2) = 1/4
2^(-3) = 1/8
2^(-4) = 1/16
etc

Since 2^t is always POSITIVE, we know that 1 + 2^t is POSITIVE, which means 1/(1 + 2^t) is POSITIVE

Does that help?

Cheers,
Brent

In some questions we have the statement n is an integer so we assume that it is positive or negative?
greenlight-admin's picture

If n is an integer, then n can be negative, positive, or zero (which is neither positive nor negative)

Here are all the integers: {. . . . -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, . . . . }

Cheers
Brent

Hi,in the following case are able to susbtract X in case both bases are the same?
https://greprepclub.com/forum/x-is-a-negative-integer-1895.html
greenlight-admin's picture

Link: https://greprepclub.com/forum/x-is-a-negative-integer-1895.html

Sorry, can you elaborate? What part of the solution are you referring to?

I meant the exponent

3 to the power of x+1 and 2 to the power of x

in both cases subtract -x so I will have 3 and 2,

will it be possible in the case that X is a positive integer?
greenlight-admin's picture

No, that strategy won't work.

It's a perfectly valid strategy to subtract the SAME VALUE from each quantity, however that's not what you're going here.

Consider this example:
QUANTITY A: 3²
QUANTITY B: 5²

If we subtract the exponent 2 in both quantities, we aren't subtracting the SAME value from both quantities.
QUANTITY A becomes 3, which means we subtracted 6 from quantity A
QUANTITY B becomes 5, which means we subtracted 20 from quantity B

Does that help?

Cheers,
Brent

In this case
https://greprepclub.com/forum/t-is-an-integer-9385.html

Given:
Quantity A: 1/(1 + 2^t)
Quantity B: 1/(1 + 3^t)

Since 2^t is POSITIVE for all integer values of t, we know that 1 + 2^t is also POSITIVE
This means we can safely multiply both quantities by (1 + 2^t) but in the case that instead of +2^t we have -2^t, we will not be able to multiply both quantities by (1 - 2^t)? because it will lead to different answers?
Thank you


greenlight-admin's picture

Solution link: https://greprepclub.com/forum/t-is-an-integer-9385.html#p19958
Yes, that's correct. Multiplying both quantities by a negative value will lead to a different answer.

Hi Brent,

https://greprepclub.com/forum/if-5n-4n-x-8511.html
I tried solving this question using this approach:
5n/4n-x=1/0.788
= 0.788*5n=4n-x
= (788/1000)5n=4n-x
= 788n=200(4n-x)
= 788n=800n-200x
= -12n=-200x
= x/n=50/3

I am getting 50/3 instead of 3/50. Could you please point out where I am wrong.

Thanks,
Ketan
greenlight-admin's picture

You're soooooo close!!

Everything is correct to the point where you write: -12n = -200x
Now, divide both sides by n to get: -12 = -200x/n
Divide both sides by -200 to get: -12/-200 = x/n
Simplify to get: 3/50 = x/n

Cheers,
Brent

I've noticed that questions like: https://greprepclub.com/forum/x-y-and-p-are-integers-and-xyp-0-if-14322.html is one of my weak areas. I waver between trying numbers or reasoning through the question to save time but have had little luck. What if the second set of numbers you tried resulted in III being true? would you have tried another set of numbers? Isn't that time expensive?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/x-y-and-p-are-integers-and-xyp-0-if-14322....

This is a super hard question (at present, only 25% of users have correctly answered it, which is VERY low when you consider the fact that, if everybody randomly guessed A, B, C, D or E, the success rate would be 20%)

If, for statement III, I test two sets of numbers, and the statement was true in both cases, I'd feel better about the prospects of statement II I being true, but I'd likely test 1 or 2 more cases first.

Cheers,
Brent

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