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Given p=|q| and \(q=3^a-(27^{40}+27^8)\), which of the following values of 'a' yields the least value for p ?

(A) 120
(B) 140
(C) 180
(D) 185
(E) 190

since p is always positive and p = |q|, we have to find the least |q|
now \(q=3^a-(27^{40}+27^8)=3^a-27^8(27^{32}+1)=3^a-27^8(27^{32}=3^a-27^{40}=3^a-3^{120}\),
so a ~ 120

A

what happened to the +1 after the 27^{32}?
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Given p=|q| and \(q=3^a-(27^{40}+27^8)\), which of the following values of 'a' yields the least value for p ?

(A) 120
(B) 140
(C) 180
(D) 185
(E) 190

since p is always positive and p = |q|, we have to find the least |q|
now \(q=3^a-(27^{40}+27^8)=3^a-27^8(27^{32}+1)=3^a-27^8(27^{32}=3^a-27^{40}=3^a-3^{120}\),
so a ~ 120

A

what happened to the +1 after the 27^{32}?

Hi....
Since 1 is very very small as compared to 27^{32}, we can discard 1 as we are looking at an approximate value
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Bunuel pushpitkc Bunuel niks18 gmatbusters KarishmaB

I have a small query regarding the penultimate step by chetan2u as below:
Quote:
\(3^a-3^{120}\),
so a ~ 120

Here is the question stem again which asks for a definite value and not approximation.

Quote:
Given p=|q| and \(q=3^a-(27^{40}+27^8)\), which of the following values of 'a' yields the least value for p ?

I believe chetan2u took the highlighted text to infer that \(3^a\) must be a positive value.

What I did not get was: if a is actually equal to 120 then the final value of q will be 0 and p will be 0 , which voilates Q stem.
Can you suggest why we did not take next positive value i.e. 140 as correct OA?
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Hi

Query1:

I believe chetan2u took the highlighted text to infer that \(3^a\) must be a positive value.. it is not to be inferred from the question stem.

3^a is always positive- 3^a can never be negative and 3^a tends to zero only if a approaches negative infinity.

Concept: (positive)^x, is always positive, it can not be negative. (positive)^x tends to zero only if a approaches negative infinity.



Query2: if a is actually equal to 120 then the final value of q will be 0 and p will be 0 , which violates Q stem
I am afraid, I couldn't understand this point. How it violates the Question stem.- please explain the query further to let me expand.

In fact , p=|q|. Hence the minimum value of p can be equal to zero.( it cannot be negative). So , as explained in above posts, p approaches zero, when a is 120. (out of the given 5 options).


adkikani
gmatbusters

I have a small query regarding the penultimate step by chetan2u as below:
Quote:
\(3^a-3^{120}\),
so a ~ 120

Here is the question stem again which asks for a definite value and not approximation.

Quote:
Given p=|q| and \(q=3^a-(27^{40}+27^8)\), which of the following values of 'a' yields the least value for p ?

I believe chetan2u took the highlighted text to infer that \(3^a\) must be a positive value.

What I did not get was: if a is actually equal to 120 then the final value of q will be 0 and p will be 0 , which voilates Q stem.
Can you suggest why we did not take next positive value i.e. 140 as correct OA?
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adkikani
Bunuel pushpitkc Bunuel niks18 gmatbusters KarishmaB

I have a small query regarding the penultimate step by chetan2u as below:
Quote:
\(3^a-3^{120}\),
so a ~ 120

Here is the question stem again which asks for a definite value and not approximation.

Quote:
Given p=|q| and \(q=3^a-(27^{40}+27^8)\), which of the following values of 'a' yields the least value for p ?

I believe chetan2u took the highlighted text to infer that \(3^a\) must be a positive value.

What I did not get was: if a is actually equal to 120 then the final value of q will be 0 and p will be 0 , which violates Q stem.
Can you suggest why we did not take next positive value i.e. 140 as correct OA?

Hi adkikani,

As you know, |q|=q when q>0
|q|=-q when q<0
So, p=|q|=positive
Hence, determining the least value of 'p' is same as determining the least value(minimum) of q.

Given \(q=3^a-(27^{40}+27^8)=3^a-(3^{120}-3^{24})=3^a-3^{120}-3^{24}\)--------------(1)

adkikani
What I did not get was: if a is actually equal to 120 then the final value of q will be 0 and p will be 0 , which violates Q stem.

when a=120, \(q=3^{120}-3^{120}+3^{24}\)=\(-{3^{24}}\neq0\)

Quote:
\(3^a-3^{120}\),
so a ~ 120
Here is the question stem again which asks for a definite value and not approximation.

Here a=120 is not approximated. When a=120, the given expression has the least value of q, i.e. \(-3^{24}\). So, \(p=|q|=-q=-(-3^{24})=3^{24}\)

Quote:
Can you suggest why we did not take next positive value i.e. 140 as correct OA?

Let's take a=140, so (1) becomes, \(q=3^{140}-3^{120}-3^{24}\). As \(3^{140}-3^{120}\)is positive, so \(|q|> 3^{24}\)
But we want the least value. Therefore, we have to discard a=140.

Quote:
I believe chetan2u took the highlighted text to infer that \(3^a\) must be a positive value.Given p=|q| and \(q=3^a-(27^{40}+27^8)\), which of the following values of 'a' yields the least value for p ?

No, the highlighted portion signifies 'p' must be positive. (q<0 (say -5) implies p=|q|=-q=-(-5)=positive; q>0 implies p=|q|=positive)

P.S:- We can have value of the expression(|q|) less than \(3^{24}\) but we are told to determine the least value of 'p' from the given 5 answer options.(a=120, 140,180,185, and 190)
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adkikani
Bunuel pushpitkc Bunuel niks18 gmatbusters KarishmaB

I have a small query regarding the penultimate step by chetan2u as below:
Quote:
\(3^a-3^{120}\),
so a ~ 120

Here is the question stem again which asks for a definite value and not approximation.

Quote:
Given p=|q| and \(q=3^a-(27^{40}+27^8)\), which of the following values of 'a' yields the least value for p ?

I believe chetan2u took the highlighted text to infer that \(3^a\) must be a positive value.

What I did not get was: if a is actually equal to 120 then the final value of q will be 0 and p will be 0 , which voilates Q stem.
Can you suggest why we did not take next positive value i.e. 140 as correct OA?

Most of the points have been made amply clear in above posts. However, the point on approximation..
What is the least value of p, it is 0 as p=|q|
And at a=120, it is not 0, it is 3^120-(3^120+3^24)=-3^24
So p =|-3^24|=3^24

So that is not equal to 0..
But are we looking for 'a' to get the least value of p...NO then a will be 120.xyz, something in decimals.
We are looking for the least value for the FOLLOWING values
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