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Manager  Joined: 02 Aug 2007
Posts: 194
Schools: Life
If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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4
16 00:00

Difficulty:   55% (hard)

Question Stats: 74% (02:59) correct 26% (03:18) wrong based on 807 sessions

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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
Math Expert V
Joined: 02 Sep 2009
Posts: 58142
Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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9
2
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:

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

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

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Director  Joined: 14 Aug 2007
Posts: 588
Re: PS - Powers, what is the value of x?  [#permalink]

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1
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
Intern  Joined: 14 Nov 2012
Posts: 3
Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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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?
Manager  B
Joined: 25 Jun 2012
Posts: 132
Location: United States
GMAT 1: 700 Q47 V40 GMAT 2: 740 Q48 V44 GPA: 3.48
Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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3^0 is substituted for 1 to show both sides of the equation with the same base

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Intern  Joined: 07 Nov 2012
Posts: 3
Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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Can anyone explain how the 18 comes into play?

Also 2^0 in the second explanation.

Thanks.
Intern  Joined: 16 May 2013
Posts: 6
Concentration: Nonprofit, Real Estate
GRE 1: Q160 V157 GPA: 3.85
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Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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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
Math Expert V
Joined: 02 Sep 2009
Posts: 58142
Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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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]

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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:

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

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

I did in the same way, however taking more time
Any shorter approach please?
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Joined: 02 Sep 2009
Posts: 58142
Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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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:

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

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

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

Not every question has a silver bullet approach.
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Manager  Joined: 20 Dec 2013
Posts: 225
Location: India
Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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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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Senior Manager  Joined: 12 Aug 2015
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Concentration: General Management, Operations
GMAT 1: 640 Q40 V37 GMAT 2: 650 Q43 V36 GMAT 3: 600 Q47 V27 GPA: 3.3
WE: Management Consulting (Consulting)
Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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Hi

any shorter route to solution? this is a classical manhattan cat question - simple concept, a lot of iterations. gmac would unlikely post something like this, wouldnt it?

looking forward to any shortcut
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Manager  Joined: 10 Jun 2015
Posts: 113
Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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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

You have only two prime factors, 2 and 3, involved in this question of exponents.
162=2*3^4
36 =2^2 * 3^2
Therefore, 2^-2x and 2^2x get cancelled.
Now, check only exponents of 3
3^12x * 3^6 * 3^-8x * 3^2x * 3^12 * 3^-4x=1
3^2x * 3^18 = 3^0
Implies 2x+18 = 0
x = -9
Intern  Joined: 20 Aug 2016
Posts: 9
Location: United Kingdom
Schools: Stanford '19 (S)
GMAT 1: 720 Q48 V40 GPA: 4
Technique: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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1
Hi everyone - Thank you all for your posts.. I thought I would try do my bit

Powers and bases and rate questions have been my nemisis when it has come to pushing the upper ends of quant scores - mainly as I kept getting these wrong! .. I would lose track of the calculations in trying to keep the scientific notation. It was costing me 1-2 questions a test and really frustrating me but most importantly was costing time.

The below technique is one which has turned these problems on their head for me ... i hope its useful It is not perfect but its very helpful for these types of questions, especially if like me you fight the constant battle against the demon of attention to detail (and his sidekick - messy writing)

My method for base questions is analyse vertically in form Base - Power - Variable(s) - Exponent. Using this, this question came together in around 1m50 while I reckon it would have been a 3m+ expedition prior to this method.

Step 1 .... quick scan reveals that this is likely to be a base question. Sketch quick table and then do the factorisation one by one.

27 - 3^3
162 = 81 x 2 = 3^4 and 2^1
36 = 9 x 4 = 3^2 and 2^2
9 = 3^2 Powers of 2 cancel
Base 3 / Exponent = 12x – 8x + 2x – 4x + 12 + 6 = 18 +2x
3^(18+2x) = 3^0
X= -9
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Posts: 2523
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30 GPA: 3.92
WE: General Management (Transportation)
Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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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

needs to be used math function on the forum...it gets difficult to understand what is what...

27 = 3^3
27^(4x + 2) = 3^(12x+6)

162 = 3^4 * 2
162^-2x = 3^(-8x) * 2^(-2x)

36^x = 3^2x * 2^2x

9^(6 – 2x) = 3^(12-4x)

now.

3^(12x+6) * 3^(-8x) * 2^(-2x) * 3^2x * 2^2x * 3^(12-4x)

powers of 2 cancel, and we have 2^0, which is equal to 1.
rest, we can rewrite as 3 to a long power:
12x+6-8x+2x+12-4x=2x+18.
since 3^(2x+18) = 1, it must be true that 2x+18=0. x=-9
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Schools: Yale '18
GMAT 1: 650 Q43 V37 GRE 1: Q157 V158 GPA: 2.66
Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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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

No shortcuts in this problem- best to just think fast and work hard to force out the answer

74x + 2 × 162-2x × 36x × 96 – 2x = 1
(33)4x + 2 × (2 × 34)-2x × (22 × 32)x × (32)6 – 2x = 1
312x + 6 × 2-2x × 3-8x × 22x × 32x × 312 – 4x = 1
2-2x + 2x × 312x + 6 – 8x + 2x + 12 – 4x = 1
20 × 3 2x + 18 = 1
3 2x + 18 = 1

Any number to the zero power equals 1 so set 3's algebraic expression exponent equal to 0

2x + 18 = 0

(-9)

The correct answer is A.
Non-Human User Joined: 09 Sep 2013
Posts: 12427
Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is  [#permalink]

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_________________ Re: If 27^(4x + 2) 162^-2x 36^x 9^(6 2x) = 1, then what is   [#permalink] 21 Jul 2018, 23:31
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