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

Question

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

Bunuel · Accepted Answer

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.

alpha_plus_gamma · Answer

(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

morfin · Answer

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?

CA88 · Answer

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

Posted from my mobile device

hdpichardo · Answer

Can anyone explain how the 18 comes into play?

Also 2^0 in the second explanation.

Thanks.

NicolasFSS · Answer

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

Bunuel · Answer

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.

PareshGmat · Answer

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

Bunuel · Answer

Not every question has a silver bullet approach.

AKG1593 · Answer

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

Posted from my mobile device

shasadou · Answer

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

mathivanan · Answer

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

kconn · Answer

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

https://i66.tinypic.com/2yv124l.png

Powers of 2 cancel
Base 3 / Exponent = 12x – 8x + 2x – 4x + 12 + 6 = 18 +2x 
3^(18+2x) = 3^0 
X= -9

mvictor · Answer

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

Nunuboy1994 · Answer

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.

Victz · Answer

Knowing the powers of 2 till 2^10 and the powers of 3 till 3^5 helps me to answer this Q in 2:50, without this knowledge, it took me close to 4 min,

tomloveless · Answer

Where did the "18+2x" come from on the last part of this question?

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

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