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Given p=|q| and q=3^a-(27^40+27^8), which of the following values of '

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Given p=|q| and q=3^a-(27^40+27^8), which of the following values of '  [#permalink]

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New post 14 Jul 2018, 07:32
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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
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Re: Given p=|q| and q=3^a-(27^40+27^8), which of the following values of '  [#permalink]

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New post 14 Jul 2018, 07:48
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PKN wrote:
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
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: Given p=|q| and q=3^a-(27^40+27^8), which of the following values of '  [#permalink]

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New post 23 Jul 2018, 17:37
chetan2u wrote:
PKN wrote:
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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Re: Given p=|q| and q=3^a-(27^40+27^8), which of the following values of '  [#permalink]

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New post 23 Jul 2018, 18:34
nati1516 wrote:
chetan2u wrote:
PKN wrote:
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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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: Given p=|q| and q=3^a-(27^40+27^8), which of the following values of '  [#permalink]

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New post 27 Jul 2018, 06:01
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Hi,

I think it could also be solved in the following way:
q=3^a−(27^40+27^8)=3^a- ((3^3)^40 + (3^3)^8

= 3^a - (3^120 + 3^24)

= 3^a - 3^120 - 3^24

if a = 120 then the result would be - 3^24. For all other values, it will be greater than this.

So, Answer is A.
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Given p=|q| and q=3^a-(27^40+27^8), which of the following values of '  [#permalink]

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New post 28 Jul 2018, 01:27
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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Re: Given p=|q| and q=3^a-(27^40+27^8), which of the following values of '  [#permalink]

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New post 28 Jul 2018, 02:56
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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 wrote:
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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Given p=|q| and q=3^a-(27^40+27^8), which of the following values of '  [#permalink]

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New post 28 Jul 2018, 03:00
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adkikani wrote:
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 wrote:
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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Re: Given p=|q| and q=3^a-(27^40+27^8), which of the following values of '  [#permalink]

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New post 28 Jul 2018, 03:45
adkikani wrote:
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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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: Given p=|q| and q=3^a-(27^40+27^8), which of the following values of ' &nbs [#permalink] 28 Jul 2018, 03:45
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