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IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:

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IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:  [#permalink]

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New post 09 Jun 2019, 19:42
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If \(P = 9^{1/2} - 9^{1/3} - 9^{1/4}\), then P is:

A) Less than 0

B) Equal to 0

C) Between 0 and 2

D) Between 2 and 3

E) Greater than 3

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Re: IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:  [#permalink]

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New post 09 Jun 2019, 22:04
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I solved it by estimating the numbers like

9^1/2=3
9^1/3 must be greater 2 as 2*2*2 is 8.
9^1/4 is 1.7 as root 3 as it can be simplified.

Therefore 3-3.7 must be less than 0 IMO option A.

But would like to know the official solution (or a more elegant/template one)

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Re: IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:  [#permalink]

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New post 15 Jun 2019, 04:36
something is wrong with the answer to this question.

using calculator:

9^1/2=3
-
9^1/3 = 1.12983
-
9^1/4 = 1.16652
=
0.70365,

Hence C.
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Re: IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:  [#permalink]

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New post 15 Jun 2019, 04:55
omavsp wrote:
something is wrong with the answer to this question.

using calculator:

9^1/2=3
-
9^1/3 = 1.12983
-
9^1/4 = 1.16652
=
0.70365,

Hence C.


You must have made some error with the calculations cause 2*2*2=8 so 9^1/3 must be greater than 2 (2.08 to be precise, checked with calculator)
9^1/4=3^1/2=1.732. Root 3 is a standard term which is 1.7

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Re: IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:  [#permalink]

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New post 15 Jun 2019, 05:10
\(P= 9^{1/4}[9^{1/4}-9^{1/12}-1]\)
\(1<9^{1/4}<2\)
\(9^{1/12}>1\)
Hence
\([9^{1/4}-9^{1/12}-1]<0\)
or
\(P= 9^{1/4}[9^{1/4}-9^{1/12}-1]\)<0

Mahmoudfawzy83 wrote:
If \(P = 9^{1/2} - 9^{1/3} - 9^{1/4}\), then P is:

A) Less than 0

B) Equal to 0

C) Between 0 and 2

D) Between 2 and 3

E) Greater than 3
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Re: IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:  [#permalink]

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New post 16 Jun 2019, 10:17
nick1816 wrote:
\(P= 9^{1/4}[9^{1/4}-9^{1/12}-1]\)
\(1<9^{1/4}<2\)
\(9^{1/12}>1\)
Hence
\([9^{1/4}-9^{1/12}-1]<0\)
or
\(P= 9^{1/4}[9^{1/4}-9^{1/12}-1]\)<0

Mahmoudfawzy83 wrote:
If \(P = 9^{1/2} - 9^{1/3} - 9^{1/4}\), then P is:

A) Less than 0

B) Equal to 0

C) Between 0 and 2

D) Between 2 and 3

E) Greater than 3



Did not understand
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Re: IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:  [#permalink]

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New post 16 Jun 2019, 12:08
9^1/2 = 3
9^ 1/4= 1.732
Let 9^1/3 will have min value as 1.732 ( assumption) , no need for exact value
together 9^1/3 and 9^1/4 is more than 3 ( Considering min value)
so 9^1/2 - ( 9^1/3 +9^1/4) = value will always be less than zero.
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IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:  [#permalink]

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New post 16 Jun 2019, 18:50
msali1983 wrote:
9^1/2 = 3
9^ 1/4= 1.732
Let 9^1/3 will have min value as 1.732 ( assumption) , no need for exact value
together 9^1/3 and 9^1/4 is more than 3 ( Considering min value)
so 9^1/2 - ( 9^1/3 +9^1/4) = value will always be less than zero.


But then what if you dont know the value of 9^1/3 and 9^1/4. I dont think that you can rely on this approach
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Re: IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:  [#permalink]

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New post 16 Jun 2019, 19:02
I did it like this:
take 9^1/4 common ,
P= 9 ^1/4 *( 9^1/4 -9^1/12 - 1)
P= 9^1/4 *( 1.~ - 1.~ - 1)
P= 9^1/4* ( overall a negative )
P= negative
hence A.
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Re: IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:  [#permalink]

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New post 18 Jun 2019, 21:43
amitcpotnis wrote:
msali1983 wrote:
9^1/2 = 3
9^ 1/4= 1.732
Let 9^1/3 will have min value as 1.732 ( assumption) , no need for exact value
together 9^1/3 and 9^1/4 is more than 3 ( Considering min value)
so 9^1/2 - ( 9^1/3 +9^1/4) = value will always be less than zero.


But then what if you dont know the value of 9^1/3 and 9^1/4. I dont think that you can rely on this approach


Here is how i solved it:
Square root, cube root, fourth root are increasing function. for eg x^(1/2) greater the value of X greater is the value of square root, cube root, fourth root.
Lets use this info in this question.

square root of 9 is 3
Cube root of 8 is 2. 9 is greater than 8 so the cube root of 9 will be greater than 2 at-least.
Fourth root of 1 is 1. 9 is greater than 1. so the fourth root of 9 will be greater than 1.
Now lets combine the above three information.
You have 3 - (something greater than 2) - (something greater than 1)

hence the answer will be less than 0.

hope the explanation helped.

Please give kudos if you like the explanation.
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IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:  [#permalink]

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New post 27 Jun 2019, 12:58
Mahmoudfawzy83 wrote:
If \(P = 9^{1/2} - 9^{1/3} - 9^{1/4}\), then P is:

A) Less than 0

B) Equal to 0

C) Between 0 and 2

D) Between 2 and 3

E) Greater than 3



Hi All,

I don't seem to understand why there is so much confusion regarding this question:

\(P = 9^{1/2} - 9^{1/3} - 9^{1/4}\),

then you should square both sides

\(P^2 = {9^{1/2}}^{2} - {9^{1/3}}^{2} - {9^{1/4}}^{2}\),

\(P^2 = 9 - {9^{2}}^{1/3} - 9^{1/2}\),

\(P^2 = 9 - 3 - 3\)

Therefore \(p = 3^{1/2}\)
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IF P = 9^{1/2} - 9^{1/3} - 9^{1/4}, then P is:   [#permalink] 27 Jun 2019, 12:58
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