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Math Expert V
Joined: 02 Sep 2009
Posts: 56300
If 4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^24, what is the value of x?  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 74% (01:32) correct 26% (01:24) wrong based on 214 sessions

### HideShow timer Statistics If $$4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^{24}$$, what is the value of x?

(A) 3
(B) 5
(C) 6
(D) 8.5
(E) 11.5

Kudos for a correct solution.

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Manager  Joined: 29 Jul 2015
Posts: 157
Re: If 4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^24, what is the value of x?  [#permalink]

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Bunuel wrote:
If $$4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^{24}$$, what is the value of x?

(A) 3
(B) 5
(C) 6
(D) 8.5
(E) 11.5

Kudos for a correct solution.

$$4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^{24}$$
or $$4^{2x} + 4^{2x} + 4^{2x} + 4^{2x} = 4^{24}$$
or $$4(4^{2x}) = 4^{24}$$
or $$4^{2x+1} = 4^{24}$$

or $$2x+1 = 24$$
or $$2x = 23$$
or $$x = 11.5$$

Manager  Joined: 09 Aug 2015
Posts: 84
GMAT 1: 770 Q51 V44 GPA: 2.3
If 4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^24, what is the value of x?  [#permalink]

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1
Make all the bases common, in this case you can choose 2 or 4.

I chose 2.

Then the equation becomes
$$2^{4x} + 2^{4x} + 2^{4x} + 2^{4x} = 2^{48}$$
Factor out $$2^{4x}$$ from left side, which becomes
$$2^{4x}(1+1+1+1) = 2^{48}$$
$$2^{4x}(4) = 2^{48}$$

Since 4 is just $$2^{2}$$and we can always add exponents of the same base in a product, left side can be rewritten as
$$2^{4x + 2}$$

So now $$4x + 2 = 48$$
$$4x = 46.$$

At this point you dont have to continue, see that the closest answers are 11.5 and 8. $$4*8$$ is clearly less than 46 => Answer must be E
Director  P
Joined: 21 May 2013
Posts: 655
Re: If 4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^24, what is the value of x?  [#permalink]

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1
Bunuel wrote:
If $$4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^{24}$$, what is the value of x?

(A) 3
(B) 5
(C) 6
(D) 8.5
(E) 11.5

Kudos for a correct solution.

Now, 4^2x can be written as 2^2(2x)=2^4x
Therefore, (4)2^4x=2^48
2^4x+2=2^48
4x+2=48
4x=46
x=11.5
Intern  Joined: 12 Nov 2013
Posts: 40
Re: If 4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^24, what is the value of x?  [#permalink]

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1

Converting to common base of 2, we get 2^4x + 2^4x + 2^4x + 2^4x = 2^48

> Take 2^4x common: 2^4x (1+1+1+1) = 2^48

>2^4x (4) = 2^48

> 2^4x +2 = 2^48

> 4x + 2 = 48

> 4x = 46

> x = 11.5
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Math Expert V
Joined: 02 Sep 2009
Posts: 56300
Re: If 4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^24, what is the value of x?  [#permalink]

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Bunuel wrote:
If $$4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^{24}$$, what is the value of x?

(A) 3
(B) 5
(C) 6
(D) 8.5
(E) 11.5

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

Here, you can note that you do a few things quite well with exponential terms:

Break their bases down to primes to get common bases. Multiply them.

So when you see a problem like this, you should recognize your strengths with exponents and look to rearrange the algebra to take advantage of them. Breaking the 4 terms down to prime factors (2), you get:

$$(2^2)^{2x} + 2^{4x} + (2^2)^{2x} + 2^{4x} = (2^2)^{24}$$

Then you can get back to multiplication to eliminate the parentheses:

$$2^{4x} + 2^{4x} + 2^{4x} + 2^{4x} = 2^{48}$$

Again, look for chances to do what you do well – and you know that if you can multiply the terms on the left instead of adding them, you’re then multiplying exponential terms with a common base…that’s your strength. In this problem, you may recognize quickly that you have four of the same term, and can express it as:

$$4(2^{4x}) = 2^{48}$$

Were the problem slightly more difficult, or you didn’t make that recognition, you might need to factor out the common exponential term so that you can multiply it that way:

$$2^{4x}(1+1+1+1)=2^{48}$$

$$2^{4x}(4) = 2^{48}$$

Either way, you end up with the same multiplication, which is what’s most important – now you’re doing what you do well.

$$4(2^{4x}) = 2^{48}$$

One more step is to, again, break down different bases into primes so that you can again multiply exponents. 4 = 2^2, so you have:

$$2^2(2^{4x}) = 2^{48}$$

And because you’re pretty quick when multiplying exponents of the same base, you should recognize that that can be expressed as:

$$2^{4x+2} = 2^{48}$$

Now that the bases are the same and the terms are set equal, you can note that:

$$4x+2 = 48$$

$$4x = 46$$

$$x = 11.5$$, and the answer is E.
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Re: If 4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^24, what is the value of x?  [#permalink]

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Bunuel wrote:
If $$4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^{24}$$, what is the value of x?

(A) 3
(B) 5
(C) 6
(D) 8.5
(E) 11.5

Kudos for a correct solution.

$$4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^{24}$$

Or, $$2^{4x} + 2^{4x} + 2^{4x} + 2^{4x} = 2^{48}$$

Or, $$2^{4x} ( 1 + 1 + 1 + 1 ) = 2^{48}$$

Or, $$2^{4x} *2^2 = 2^{48}$$

Or, $$2^{4x} = 2^{46}$$

Or, $$4x = 48$$

Or, $$x = 11.5$$

Thus, answer must be (E) 11.50
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Re: If 4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^24, what is the value of x?  [#permalink]

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Bunuel wrote:
If $$4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^{24}$$, what is the value of x?

(A) 3
(B) 5
(C) 6
(D) 8.5
(E) 11.5

Let’s simplify the given equation:

4^2x + 2^4x + 4^2x + 2^4x = 4^24

2^4x + 2^4x + 2^4x + 2^4x = 2^48

2^4x(1 + 1 + 1 + 1) = 2^48

2^4x(4) = 2^48

2^4x(2^2) = 2^48

2^(4x + 2) = 2^48

4x + 2 = 48

4x = 46

x = 46/4 = 23/2 = 11.5

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Re: If 4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^24, what is the value of x?  [#permalink]

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First Solution post on GMAT Club
Exited
Simplify and get
2^4x(2^2) = 2^48
Simplify this and get
4x+2 = 48
Simplify this and get 11.5

Bunuel wrote:
If $$4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^{24}$$, what is the value of x?

(A) 3
(B) 5
(C) 6
(D) 8.5
(E) 11.5

Kudos for a correct solution.

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Posts: 11709
Re: If 4^{2x} + 2^{4x} + 4^{2x} + 2^{4x} = 4^24, what is the value of x?  [#permalink]

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