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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