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If \(\mathrm{x}\) is a real number and \(4^{(4^{4})} = 2^{(2^\mathrm{x})}\), what is the value of \(\mathrm{x}\) ?
We have the equation \(4^{(4^4)} = 2^{(2^x)}\) and we need to find the value of x.
Let me break this down in everyday terms: We're looking for a number x such that when we plug it into the expression \(2^{(2^x)}\), we get the same result as \(4^{(4^4)}\).
Think of it like this - we have two different-looking expressions that give us the same final number. Our job is to figure out what x must be to make this work.
Process Skill: TRANSLATE - Converting the equation into a clear mathematical goal
Since we're dealing with powers of both 2 and 4, and we know that \(4 = 2^2\), let's express everything using base 2. This is like converting different currencies to the same currency so we can compare them properly.
Why base 2? Because both 2 and 4 are powers of 2, so this will make our comparison much cleaner.
The right side is already in base 2: \(2^{(2^x)}\)
For the left side, we need to rewrite \(4^{(4^4)}\) using base 2.
Since \(4 = 2^2\), we can write: \(4^{(4^4)} = (2^2)^{(4^4)}\)
Let's work through this systematically:
First, let's calculate \(4^4\):
\(4^4 = 4 \times 4 \times 4 \times 4 = 16 \times 16 = 256\)
So our left side becomes: \(4^{(4^4)} = 4^{256}\)
Now we can rewrite this in base 2:
\(4^{256} = (2^2)^{256}\)
Using the power rule (when we raise a power to a power, we multiply the exponents):
\((2^2)^{256} = 2^{(2 \times 256)} = 2^{512}\)
So our equation now looks like: \(2^{512} = 2^{(2^x)}\)
Process Skill: SIMPLIFY - Breaking down complex expressions into manageable pieces
Now we have both sides in the same base: \(2^{512} = 2^{(2^x)}\)
When two expressions with the same base are equal, their exponents must be equal. This means:
\(512 = 2^x\)
Now we need to figure out what power of 2 gives us 512.
Let's think about powers of 2:
\(2^1 = 2\)
\(2^2 = 4\)
\(2^3 = 8\)
\(2^4 = 16\)
\(2^5 = 32\)
\(2^6 = 64\)
\(2^7 = 128\)
\(2^8 = 256\)
\(2^9 = 512\)
So \(2^9 = 512\), which means \(x = 9\).
The value of x is 9.
Let's verify: If \(x = 9\), then \(2^x = 2^9 = 512\), so \(2^{(2^x)} = 2^{512}\).
We showed that \(4^{(4^4)} = 2^{512}\).
Therefore, \(4^{(4^4)} = 2^{(2^x)}\) when \(x = 9\).
The answer is C.
Students often get confused by the nested exponents and may incorrectly interpret \(4^{(4^4)}\) as \((4^4)^4\) instead of \(4^{(4^4)}\). This is a critical error because \((4^4)^4 = 4^{16}\), which is vastly different from \(4^{(4^4)} = 4^{256}\). The parentheses in exponential notation indicate the order of operations - we must calculate the inner exponent first.
Students might attempt to convert everything to base 4 instead of base 2, or might try to work with the original bases without conversion. Since \(4 = 2^2\), converting to base 2 makes the problem much more manageable. Working with base 4 would make the right side (\(2^{(2^x)}\)) more complex to handle.
When calculating \(4^4 = 4 \times 4 \times 4 \times 4\), students commonly make errors like getting 64 (which is \(4^3\)) or 1024 (which is \(4^5\)) instead of the correct answer 256. This error propagates through the entire solution since the final exponent depends on this calculation.
When converting \((2^2)^{256}\) to \(2^{(2\times256)}\), students might add the exponents instead of multiplying them, getting \(2^{(2+256)} = 2^{258}\) instead of \(2^{512}\). This is a fundamental error with the power rule: \((a^m)^n = a^{(mn)}\), not \(a^{(m+n)}\).
When determining what power of 2 equals 512, students often make mistakes in the sequence of powers of 2. Common errors include confusing \(2^8 = 256\) with \(2^9 = 512\), or stopping the calculation too early and not reaching \(2^9\).
Students might select 256 (which is \(4^4\)) or 512 (which is \(2^x\)) as their final answer instead of \(x = 9\). They may lose track of what the question is actually asking for - the value of x, not the value of the exponents in the intermediate steps.