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# If 3^x - 3^(x-1) = 162, then x(x - 1) =

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Intern
Joined: 09 Jan 2009
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If 3^x - 3^(x-1) = 162, then x(x - 1) = [#permalink]

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14 Jan 2009, 02:54
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If 3^x - 3^(x-1) = 162, then x(x - 1) =

A. 12
B. 16
C. 20
D. 30
E. 81

[Reveal] Spoiler:
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[Reveal] Spoiler: OA

Last edited by Bunuel on 02 Jul 2015, 13:34, edited 2 times in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.
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Re: If 3^x - 3^(x-1) = 162, then x(x - 1) = [#permalink]

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14 Jan 2009, 05:40
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Just as you'd factor out the term with the smaller power if you saw something like $$x^8 - x^7$$ (you'd factor out $$x^7$$) you can factor out the term with the smaller power here:

\begin{align*} 3^x - 3^{x-1} &= 162 \\ 3^{x-1}(3 - 1) &= 162 \\ 3^{x-1} &= 81 \\ x - 1 &= 4 \end{align*}

So x = 5, and x*(x-1) = 20.
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Intern
Joined: 09 Jan 2009
Posts: 7
Re: If 3^x - 3^(x-1) = 162, then x(x - 1) = [#permalink]

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14 Jan 2009, 11:21
Many Thanks....good answer
Manager
Joined: 14 Dec 2011
Posts: 61
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If 3^x - 3^(x-1) = 162 then x(x-1) = [#permalink]

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14 Dec 2011, 04:55
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If 3^x - 3^(x-1) = 162 then x(x-1) =

12
16
20
30
81
Senior Manager
Joined: 28 Jun 2009
Posts: 438
Location: United States (MA)
Re: If 3^x - 3^(x-1) = 162 then x(x-1) = [#permalink]

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14 Dec 2011, 07:09
For me, fastest method was plugging in the answer choices.

From the answer choices, only 12, 20 and 30 satisfy x(x-1).

Among these only 20 i.e. x = 5 satisfies : 3^x - 3^(x-1) = 162

Hence answer is C.
Intern
Joined: 25 Nov 2011
Posts: 10
Re: If 3^x - 3^(x-1) = 162 then x(x-1) = [#permalink]

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14 Dec 2011, 09:36
3^x - 3^(x-1) = 162
(3^(x-1))*(3^(x-(x-1)) - 3^((x-1)-(x-1)) = 162
(3^(x-1))*(3^1 - 3^0) = 162
(3^(x-1))*2 = (3^4)*2
(3^(x-1)) = (3^4)
x-1=4
x(x-1)=20
Manager
Joined: 09 Nov 2011
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Re: If 3^x - 3^(x-1) = 162 then x(x-1) = [#permalink]

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14 Dec 2011, 09:42
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3^x - 3^x-1=162
3^x[1-1/3]=162
3^x[2/3]=162
3^x=81*3
=>x=5
x(x-1) =5*4 = 20
Hence C
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Re: If 3^x - 3^(x-1) = 162 then x(x-1) = [#permalink]

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14 Dec 2011, 12:15
Lars,

It seems like you were really close but you solved for x - 1 instead. I am curious - how did you reason through the question? Perhaps you figured the answer has to be in the form of x(x -1), e.g. (3)(4) = 12. Even then, had you figured x has to be a little bigger (plugging in 4 for x gives you 81 which is too low), So when you reasoned the answer to be 4 you may have plugged the 4 in the (x - 1) exponent place.

Again, a quick plugging in should get you (C) 20 as (E), the only other answer with consecutive integers as factors is far too big.
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Re: If 3^x - 3^(x-1) = 162, then x(x - 1) = [#permalink]

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02 Jul 2015, 13:31
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Re: If 3^x - 3^(x-1) = 162 then x(x-1) = [#permalink]

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22 Feb 2017, 11:28
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Re: If 3^x - 3^(x-1) = 162, then x(x - 1) = [#permalink]

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22 Feb 2017, 14:09
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Re: If 3^x - 3^(x-1) = 162, then x(x - 1) = [#permalink]

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24 Aug 2017, 00:37
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troop2118 wrote:
If 3^x - 3^(x-1) = 162, then x(x - 1) =

A. 12
B. 16
C. 20
D. 30
E. 81

[Reveal] Spoiler:
Attachment:
Picture 8.png

$$3^x - 3^{(x-1)} = 162$$

$$3^x - (3^x*3^{-1}) = 162$$

$$3^x - (\frac{3^x}{3^{1}}) = 3^4*2$$

$$\frac{3*3^x - 3^x}{3} = 3^4*2$$

$$\frac{3^x(3 - 1)}{3} = 3^4*2$$

$$3^x(3 - 1) = 3^4*2*3$$

$$3^x(2) = 3^5*2$$

$$3^x = 3^5$$

$$x = 5$$

$$x (x-1) = 5(5-1) = 5*4 = 20$$

Answer (C)...

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Re: If 3^x - 3^(x-1) = 162, then x(x - 1) = [#permalink]

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29 Aug 2017, 16:17
troop2118 wrote:
If 3^x - 3^(x-1) = 162, then x(x - 1) =

A. 12
B. 16
C. 20
D. 30
E. 81

We can simplify the left side of the equation and then factor 162 as 3^4 * 2^1. Then we have:

3^x - 3^x * 3^-1 = 3^4 * 2^1

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

On the left side, note that the expression (1 - 3^-1) = 1 - (⅓) = ⅔. We now have:

3^x(2/3) = 3^4 * 2^1

3^x = (3^4 * 2)(3/2)

3^x = 3^4 *3

3^x = 3^5

x = 5

So x(x-1) = 5(4) = 20.

Alternate Solution:

Note that 3^x = 3 * 3^(x - 1). Then:

3^x - 3^(x - 1) = 3 * 3^(x - 1) - 3^(x - 1)

Let’s factor the common 3^(x - 1):

3 * 3^(x - 1) - 3^(x - 1) = 3^(x - 1)(3 - 1) = 3^(x - 1)(2) = 162

3^(x - 1) = 81 = 3^4

x - 1 = 4

x = 5

Then, x(x - 1) = 20.

Answer: C
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Re: If 3^x - 3^(x-1) = 162, then x(x - 1) = [#permalink]

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14 Sep 2017, 15:12
Expert's post
Top Contributor
troop2118 wrote:
If 3^x - 3^(x-1) = 162, then x(x - 1) =

A. 12
B. 16
C. 20
D. 30
E. 81

[Reveal] Spoiler:
Attachment:
Picture 8.png

Given: 3^x - 3^(x-1) = 162
Factor to get: [3^(x-1)][3^1 - 1] = 162
Simplify to get: [3^(x-1)][2] = 162
Divide both sides by 2 to get: 3^(x-1) = 81
Rewrite the right side as 3^(x-1) = 3^4
So, x - 1 = 4
This means x = 5

We get x(x - 1) = (5)(5 - 1) = (5)(4) = 20

Answer: C

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Re: If 3^x - 3^(x-1) = 162, then x(x - 1) =   [#permalink] 14 Sep 2017, 15:12
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