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If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is

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If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is [#permalink] New post 10 Jul 2008, 18:05
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If 27^(4x + 2) × 162^-2x × 36^x × 9^(6 – 2x) = 1, then what is the value of x?

A. -9
B. -6
C. 3
D. 6
E. 9
[Reveal] Spoiler: OA
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Re: PS - Powers, what is the value of x? [#permalink] New post 10 Jul 2008, 18:14
x-ALI-x wrote:
If 27^(4x + 2) × 162^-2x × 36^x × 9^(6 – 2x) = 1, then what is the value of x?


A) -9
B) -6
C) 3
D) 6
E) 9


(3^3)^(4x + 2) * (3^4 * 2)^-2x * (3^2 * 2^2) ^x * (3^2)^(6-2x) = 1

the 2 ^ -2x from second term and 2^2x from the third term gets canclled

giving us all all terms with base 3.

3^ (2x + 18) = 1

2x + 18 must be 0 in order to satisfy this

x= -9

A
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Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is [#permalink] New post 14 Nov 2012, 12:03
This is the last part of the answer from the CAT explanation.

2^0 × 3^2x + 18 = 1
3^2x + 18 = 1
3^2x + 18 = 3^0
2x + 18 = 0
2x = -18
x = -9

where does the 3^0 comes from?
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Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is [#permalink] New post 14 Nov 2012, 12:06
3^0 is substituted for 1 to show both sides of the equation with the same base

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Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is [#permalink] New post 04 Jan 2013, 04:10
Can anyone explain how the 18 comes into play?

Also 2^0 in the second explanation.

Thanks.
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Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is [#permalink] New post 04 Jan 2013, 05:08
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xALIx wrote:
If 27^(4x + 2) × 162^-2x × 36^x × 9^(6 – 2x) = 1, then what is the value of x?

A. -9
B. -6
C. 3
D. 6
E. 9


27^{4x + 2} * 162^{-2x} * 36^x * 9^{6-2x} = 1

3^{3*(4x + 2)} * (2^{-2x}*81^{-2x}) * (4^{x}*9^x) * 3^{2(6-2x)} = 1

3^{3*(4x + 2)} * (2^{-2x}*3^{-8x}) * (2^{2x} *3^{2x})* 3^{2(6-2x)} = 1 --> 2^{-2x}*2^{2x}=1. So, we have that:

3^{3*(4x + 2)-8x+2x+2(6-2x)} = 1

3^{2x+18}=1 --> 2x+18=0 --> x=-9.

Answer: A.
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Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is [#permalink] New post 16 May 2013, 11:38
Quick question: How I am I supposed to know that 162 equals 2*3^4 ??? Is there a calculation to it or is 81= 3^4 common knowledge? thanks
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Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is [#permalink] New post 16 May 2013, 22:58
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NicolasFSS wrote:
Quick question: How I am I supposed to know that 162 equals 2*3^4 ??? Is there a calculation to it or is 81= 3^4 common knowledge? thanks


If you don't know that 81=3^4 you can do the following: 162=2*81=2*9*9=2*3^2*3^2=2*3^4.

I'd advice to know the powers of 2 till 2^10 and the powers of 3 till 3^5.
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Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is [#permalink] New post 27 Feb 2014, 00:03
Bunuel wrote:
xALIx wrote:
If 27^(4x + 2) × 162^-2x × 36^x × 9^(6 – 2x) = 1, then what is the value of x?

A. -9
B. -6
C. 3
D. 6
E. 9


27^{4x + 2} * 162^{-2x} * 36^x * 9^{6-2x} = 1

3^{3*(4x + 2)} * (2^{-2x}*81^{-2x}) * (4^{x}*9^x) * 3^{2(6-2x)} = 1

3^{3*(4x + 2)} * (2^{-2x}*3^{-8x}) * (2^{2x} *3^{2x})* 3^{2(6-2x)} = 1 --> 2^{-2x}*2^{2x}=1. So, we have that:

3^{3*(4x + 2)-8x+2x+2(6-2x)} = 1

3^{2x+18}=1 --> 2x+18=0 --> x=-9.

Answer: A.



I did in the same way, however taking more time
Any shorter approach please?
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Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is [#permalink] New post 27 Feb 2014, 04:30
Expert's post
PareshGmat wrote:
Bunuel wrote:
xALIx wrote:
If 27^(4x + 2) × 162^-2x × 36^x × 9^(6 – 2x) = 1, then what is the value of x?

A. -9
B. -6
C. 3
D. 6
E. 9


27^{4x + 2} * 162^{-2x} * 36^x * 9^{6-2x} = 1

3^{3*(4x + 2)} * (2^{-2x}*81^{-2x}) * (4^{x}*9^x) * 3^{2(6-2x)} = 1

3^{3*(4x + 2)} * (2^{-2x}*3^{-8x}) * (2^{2x} *3^{2x})* 3^{2(6-2x)} = 1 --> 2^{-2x}*2^{2x}=1. So, we have that:

3^{3*(4x + 2)-8x+2x+2(6-2x)} = 1

3^{2x+18}=1 --> 2x+18=0 --> x=-9.

Answer: A.



I did in the same way, however taking more time
Any shorter approach please?


Not every question has a silver bullet approach.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is [#permalink] New post 08 Mar 2014, 19:38
Option A.
All terms are variants of power of 3.
We reduce each term to some power of 3 and after cancellation get 3^(2x+18)=1
2x+18=0
X=-9

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Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is   [#permalink] 08 Mar 2014, 19:38
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