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

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

If you're not an expert at tougher exponent rule questions such as this, you can sometimes get to the answer with a bit of "brute force", even without knowing the exact exponent rules involved in the prompt. Here's how:

This question involves 3 raised to different "powers"; you can calculate them rather easily...

3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243

3^X and 3^(X-1) are consecutive powers of 3.
We're told that 3^X - 3^(X-1) = 162, so we just need to find 2 consecutive multiples of 3 that differ by 162.

Notice how...
3^5 - 3^4 =
243 - 81 =
162

This is EXACTLY what we're looking for.

X = 5
(X-1) = 4

So....(X)(X-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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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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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

Answer: [spoiler]C[/spoiler]

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

Final Answer:

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

Answer: C

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]
Expert Reply
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]
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.

Answer: C

Sorry in the first procedure (on lines 4 and 5) how does (3^4 * 2)(3/2) become 3^4 *3 and not 3^5/2 * 3? Doesn't the 3/2 multiply 3^4 as well and become (3^5)/2?
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Re: If 3^x - 3^(x-1) = 162, then x(x - 1) = [#permalink]
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