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If 3^a – 3^(a – 2) = 8(3^27), what is the value of 2a ?

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If 3^a – 3^(a – 2) = 8(3^27), what is the value of 2a ? [#permalink]

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New post 12 Sep 2017, 05:30
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Re: If 3^a – 3^(a – 2) = 8(3^27), what is the value of 2a ? [#permalink]

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New post 12 Sep 2017, 05:39
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Given \(3^a – 3^{(a – 2)} = 8(3^{27})\), we have been asked to find the value of 2a.

\(8(3^{27}) = (3^2 - 1)(3^27) = 3^{2+27} - 3{27}\)

Therefore we know that \(3^a – 3^{(a – 2)} = 3^{29} - 3{27} = 3^{29} - 3{(29 - 2)}\)

Therefore, a = 29 and 2a = 58(Option E)

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Re: If 3^a – 3^(a – 2) = 8(3^27), what is the value of 2a ? [#permalink]

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New post 12 Sep 2017, 07:39
Bunuel wrote:
If \(3^a – 3^{(a – 2)} = 8(3^{27})\), what is the value of 2a ?

A. 20
B. 25
C. 27
D. 29
E. 58


3^a - 3^(a-2) = 8(3^27)
3^a -(3^a/3^2) = 8(3^27)
take lcm = 3^2 and then cross multiply the denominator
(3^a)(3^2) - (3^a) = (3^2)(8)(3^27)
take (3^a) common on LHS
(3^a)[9-1] = (3^29)(8)
cancel 8 on both sides
(3^a) = (3^29)
a= 29
2a = 58
option E
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Re: If 3^a – 3^(a – 2) = 8(3^27), what is the value of 2a ? [#permalink]

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New post 12 Sep 2017, 08:29
Bunuel wrote:
If \(3^a – 3^{(a – 2)} = 8(3^{27})\), what is the value of 2a ?

A. 20
B. 25
C. 27
D. 29
E. 58


LHS: \(3^a-\frac{3^a}{3^2}\) \(= 3^a(1-\frac{1}{3^2})\) = \(3^a*\frac{8}{9}\)
or, \(3^{a-2}*8\) \(= 8*3^{27}\)
Hence \(a-2=27\) or \(a=29\). Therefore \(2a = 58\)

Option E

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Re: If 3^a – 3^(a – 2) = 8(3^27), what is the value of 2a ? [#permalink]

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New post 12 Sep 2017, 10:03
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Bunuel wrote:
If \(3^a – 3^{(a – 2)} = 8(3^{27})\), what is the value of 2a ?

A. 20
B. 25
C. 27
D. 29
E. 58


Given: \(3^a – 3^{(a – 2)} = 8(3^{27})\)

Factor \(3^{(a – 2)}\) from the left side to get: \(3^{(a – 2)}(3^2 – 1) = 8(3^{27})\)

Evaluate the part in the brackets: \(3^{(a – 2)}(8) = 8(3^{27})\)

Divide both sides by 8 to get: \(3^{(a – 2)} = 3^{27}\)

This means: \(a - 2 = 27\)

Solve: a = 29
So, 2a = (2)(29) = 58

Answer:
[Reveal] Spoiler:
E


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Re: If 3^a – 3^(a – 2) = 8(3^27), what is the value of 2a ? [#permalink]

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New post 15 Sep 2017, 10:07
Bunuel wrote:
If \(3^a – 3^{(a – 2)} = 8(3^{27})\), what is the value of 2a ?

A. 20
B. 25
C. 27
D. 29
E. 58


We can simplify the given equation:

3^a - (3^a)(3^-2) = 8(3^27)

Factor 3^a from the left side:

3^a(1 - 1/9) = 8(3^27)

3^a(8/9) = 8(3^27)

3^a = 8(3^27) x 9/8

3^a = 3^27 x 3^2

3^a = 2^29

a = 29

So, 2a = 58.

Alternate Solution:

Observe that 3^a = 3^(a - 2 + 2) = (3^(a - 2))(3^2). Then:

(3^(a - 2))(3^2) - (3^(a - 2)) = 8(3^27)

Let’s factor the common 3^(a - 2) from the left side:

3^(a - 2)(3^2 - 1) = 8(3^27)

3^(a - 2)(8) = 8(3^27)

3^(a - 2) = 3^27

a - 2 = 27

a = 29

2a = 58

Answer: E
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If 3^a – 3^(a – 2) = 8(3^27), what is the value of 2a ? [#permalink]

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New post 24 Sep 2017, 06:27
RHS
8 can be written as (9-1) or (3²-1)
8(3²⁷)=(3²-1)X(3²⁷)
=3²⁹-3²⁷
check answer choices.
Start with easy one.
let 2a=10 or a=5
plugin into LHS and check
3⁵-3³-Not correct
let 2a=58
a=29
Plugin to LHS of the equation
3²⁹-3²⁷=LHS=correct=E

Last edited by gps5441 on 24 Sep 2017, 06:30, edited 1 time in total.

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Re: If 3^a – 3^(a – 2) = 8(3^27), what is the value of 2a ? [#permalink]

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New post 24 Sep 2017, 06:29
RHS
8(3²⁷)=(3²-1)X(3²⁷)
=3²⁹-3²⁷
check answer choices.
Start with easy one.
let 2a=10 or a=5
plugin into LHS and check
3⁵-3³-Not correct
let 2a=58
a=29
Plugin to LHS of the equation
3²⁹-3²⁷=LHS=correct=E

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Re: If 3^a – 3^(a – 2) = 8(3^27), what is the value of 2a ? [#permalink]

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New post 24 Sep 2017, 07:14
Bunuel wrote:
If \(3^a – 3^{(a – 2)} = 8(3^{27})\), what is the value of 2a ?

A. 20
B. 25
C. 27
D. 29
E. 58


Or, \(3^a – 3^{(a – 2)} = 8(3^{27})\)

Or, \(3^a – \frac{3^a}{3^2} = (3^2 - 1)(3^{27})\)

Or, \(3^a*3^2 – 3^a = 3^2(3^2 - 1)(3^{27})\)

Or, \(3^a(3^2 – 1) = 3^2(3^2 - 1)(3^{27})\)

Or, \(3^a = 3^2(3^{27})\)

Or, \(3^a = 3^{29}\)

Or, \(a = 29\)

So, 2a = 58 , answer will be (E)

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Re: If 3^a – 3^(a – 2) = 8(3^27), what is the value of 2a ?   [#permalink] 24 Sep 2017, 07:14
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