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# If 4^a + 4^(a+1) = 4^(a+2) - 176, what is the value of a?

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Math Expert
Joined: 02 Sep 2009
Posts: 52343
If 4^a + 4^(a+1) = 4^(a+2) - 176, what is the value of a?  [#permalink]

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21 Jul 2016, 23:07
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35% (medium)

Question Stats:

78% (02:11) correct 22% (02:22) wrong based on 207 sessions

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If $$4^a + 4^{(a+1)} = 4^{(a+2)} - 176$$, what is the value of a?

A. 1
B. 2
C. 3
D. 4
E. 5

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Re: If 4^a + 4^(a+1) = 4^(a+2) - 176, what is the value of a?  [#permalink]

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22 Jul 2016, 00:57
Bunuel wrote:
If 4^a + 4^(a+1) = 4^(a+2) - 176, what is the value of a?

A. 1
B. 2
C. 3
D. 4
E. 5

Check directly by substituting the value.

Sub a = 2 from option B and only this satisfies the condition 16 + 64 = 256 -176 => 80 = 80.

IMO option B is correct option.

OA please...will correct if I missed anything...
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Re: If 4^a + 4^(a+1) = 4^(a+2) - 176, what is the value of a?  [#permalink]

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22 Jul 2016, 01:36
Well, I worked it out by doing as follows...
4^a+4^a×4=4^a×4^2-176
4^a×(1+4-16)=-176
4^a×(-11)=-11×16
Thus, 4^a=16
So, a=2
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Re: If 4^a + 4^(a+1) = 4^(a+2) - 176, what is the value of a?  [#permalink]

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22 Jul 2016, 01:51
4^a+4*4^a=16*4^a-11*4^2
5*4^a=16*4^a-11*4^2
11*4^2=11*4^a
2=a
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Re: If 4^a + 4^(a+1) = 4^(a+2) - 176, what is the value of a?  [#permalink]

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29 Mar 2018, 11:50
Hi All,

While there are several ways to approach this type of question, this prompt can be solved rather quickly by TESTing THE ANSWERS. Before beginning the 'math', consider this:

We know that 4^a, 4^(a+1) and 4^(a+2) are consecutive "powers of 4", so we could just "map out the possibilities and find the one that fits:

4^0 = 1
4^1 = 4
4^2 = 16
4^3 = 64
4^4 = 256
4^5 = 1024

Now, which 3 consecutive "powers of 4" fit the given equation (hint: the "-176" is a specific value)?

It's got to be 2, 3 and 4, so A = 2

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Re: If 4^a + 4^(a+1) = 4^(a+2) - 176, what is the value of a?  [#permalink]

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27 Jun 2018, 04:14
Bunuel wrote:
If $$4^a + 4^{(a+1)} = 4^{(a+2)} - 176$$, what is the value of a?

A. 1
B. 2
C. 3
D. 4
E. 5

Hi Bunuel , will you please show a solution without Testing Values? None of the solutions above are in correct format. Thanks in advance.
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Re: If 4^a + 4^(a+1) = 4^(a+2) - 176, what is the value of a?  [#permalink]

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27 Jun 2018, 04:38

Solution

Given:
• $$4^a + 4^{(a+1)}=4^{(a+2)}−176$$

To find:
• The value of a

Approach and Working:
We can rewrite the equation as follows:
• $$4^a + 4^{(a+1)}=4^{(a+2)}−176$$
Or, $$4^a + 4^{(a+1)} - 4^{(a+2)} = -176$$
Or, $$4^a (1 + 4 – 4^2) = -176$$
Or, $$4^a (1 + 4 – 16) = -176$$
Or, $$4^a * (-11) = -176$$
Or, $$4^a = \frac{-176}{-11} = 16 = 4^2$$
Or, a = 2

Hence, the correct answer is option B.

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Posts: 52343
Re: If 4^a + 4^(a+1) = 4^(a+2) - 176, what is the value of a?  [#permalink]

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27 Jun 2018, 04:41
1
msurls wrote:
Bunuel wrote:
If $$4^a + 4^{(a+1)} = 4^{(a+2)} - 176$$, what is the value of a?

A. 1
B. 2
C. 3
D. 4
E. 5

Hi Bunuel , will you please show a solution without Testing Values? None of the solutions above are in correct format. Thanks in advance.

$$4^a + 4^{(a+1)} = 4^{(a+2)} - 176$$;

$$4^a + 4^{(a+1)} - 4^{(a+2)} = - 176$$;

$$4^a + 4*4^a - 4^2*4^a = - 176$$;

$$4^a(1 + 4 - 4^2) = - 176$$;

$$4^a(-11) = - 176$$;

$$4^a = 16$$;

$$4^a = 4^2$$;

a = 2.

Hope it helps.
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Re: If 4^a + 4^(a+1) = 4^(a+2) - 176, what is the value of a?  [#permalink]

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02 Jul 2018, 09:05
Bunuel wrote:
If $$4^a + 4^{(a+1)} = 4^{(a+2)} - 176$$, what is the value of a?

A. 1
B. 2
C. 3
D. 4
E. 5

We can re-express 4^(a + 1) as (4^a)(4^1), and we can re-express 4^(a + 2) as (4^a)(4^2).

Simplifying the given equation, we have:

4^a + (4^a)(4^1) = (4^a)(4^2) - 176

4^a + (4^a)(4^1) - (4^a)(4^2) = -176

4^a(1 + 4^1 - 4^2) = -176

4^a(-11) = -176

4^a = 16

4^a = 4^2

a = 2

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Re: If 4^a + 4^(a+1) = 4^(a+2) - 176, what is the value of a? &nbs [#permalink] 02 Jul 2018, 09:05
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