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

Question

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 .

kunal555 · Answer

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

Answer:- E

ggurface · Answer

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

KS15 · Answer

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
Answer E

harishbiyani8888 · Answer

The correct answer is E.

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

Bunuel · Answer

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 .

Abhishek009 · Answer

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

ScottTargetTestPrep · Answer

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

Answer: E

aayushamj · Answer

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

MathRevolution · Answer

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

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

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

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

=>  2^{4x}  * 4 =  2^{48}

=>  2^{4x}  =  \frac{2^{48} }{ 2^2}

=>  2^{4x}  =  2^{46}

Comparing bases,

=> 4x = 46

=> x = 11.5

Answer E

Kinshook · Answer

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

2^{4x} + 2^{4x} + 2^{4x} + 2^{4x} = 2^{48} 
 2^{4x+2} = 2^{48} 
4x + 2 = 48
4x = 46
x = 11.5

IMO E

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

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

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