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

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Intern
Joined: 09 Jan 2009
Posts: 7
If 3^x - 3^(x-1) = 162, then x(x - 1) =  [#permalink]

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Updated on: 02 Jul 2015, 13:34
1
9
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25% (medium)

Question Stats:

74% (01:41) correct 26% (02:02) wrong based on 219 sessions

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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

Attachment:

Picture 8.png [ 9.86 KiB | Viewed 3911 times ]

Originally posted by troop2118 on 14 Jan 2009, 02:54.
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
3
6
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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##### General Discussion
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Re: If 3^x - 3^(x-1) = 162, then x(x - 1) =  [#permalink]

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14 Jan 2009, 11:21
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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
1
If 3^x - 3^(x-1) = 162 then x(x-1) =

12
16
20
30
81
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Re: If 3^x - 3^(x-1) = 162 then x(x-1) =  [#permalink]

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14 Dec 2011, 07:09
1
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

Intern
Joined: 25 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, 09:36
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
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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
1
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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24 Aug 2017, 00:37
3
1
troop2118 wrote:
If 3^x - 3^(x-1) = 162, then x(x - 1) =

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

Attachment:
Picture 8.png

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

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

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

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

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

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

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

$$3^x = 3^5$$

$$x = 5$$

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

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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.

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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
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

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

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

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13 Nov 2018, 11:27
Hi All,

We're told that 3^(X) - 3^(X-1) = 162. We're asked for the value of then (X)(X-1). This question can be solved rather easily with a bit of 'brute force' arithmetic.

Since we're dealing with 'powers of 3', let's map out the first several values:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243

We're subtracting two consecutive powers of 3 and ending up with 162. Looking at the list so far, we have an obvious 'pair' of values that fits what we're looking for:

3^5 and 3^4
243 - 81 = 162

Thus, X = 5 and the answer to the question is (5)(5-1) = 20

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

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13 Nov 2018, 11:35
3^x - 3^x-1 = 162
Taking 3^x-1 as common factor
3^x-1 (3-1) = 3^4 (3-1)
3^x-1 = 3^4
x-1 =4
x= 5

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

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

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13 Nov 2018, 13:00
troop2118 wrote:
If 3^x - 3^(x-1) = 162, then x(x - 1) =

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

$$? = x\left( {x - 1} \right)$$

$$\left. \begin{gathered} {3^x} - {3^{x - 1}} = {3^{x - 1}}\left( {3 - 1} \right)\,\,\, \hfill \\ 162 = 2 \cdot 81 = 2 \cdot {3^4} \hfill \\ \end{gathered} \right\}\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{stem}}} \,\,\,\,\,\,\,{3^{x - 1}} = {3^4}\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{3}}\,\, \notin \,\,\left\{ {0,1, - 1} \right\}} \,\,\,\,\,\,\,x - 1 = 4\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 20$$

This solution follows the notations and rationale taught in the GMATH method.

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

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06 Feb 2019, 17:52
ScottTargetTestPrep wrote:
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.

Hello ScottTargetTestPrep !

Would you be so kind and explain to me how did you get to the following?

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

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

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

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08 Feb 2019, 11:59
So here are some more details:

3^x = 3^[x - 1 + 1] = 3^[(x - 1) + 1] = [3^(x - 1)]*[3^1] = 3*3^(x - 1)

However, that's a long and complicated way of putting things. In reality, you would just observe that given any power, you can always decrease the exponent by one and multiply with the base to get an equivalent expression. For instance, if you have 2^5 (which is 2 multiplied by itself five times), that is equal to [2^4]*2 (which is 2 multiplied by itself four times, and then multiplied by 2 once more).

In the next line, all we did was to replace 3^x by the equivalent expression of 3*3^(x - 1) which we obtained as explained above.

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Re: If 3^x - 3^(x-1) = 162, then x(x - 1) =   [#permalink] 08 Feb 2019, 11:59
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