2^(x-1) + 2^(x+1) = 320. Find the value of x.
2^(x-1) + 2^(x+1) = 320. Find the value of x.
A6
B7
C8
D5
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Correct Answer: [7]
Explanation:
We start with the equation:
\(2^{x-1} + 2^{x+1} = 320\)
Factor out \(2^{x}\) from both terms:
\(2^{x-1} = \frac{2^x}{2}\) and \(2^{x+1} = 2 \cdot 2^x\)
Substitute these into the equation:
\( \frac{2^x}{2} + 2 \cdot 2^x = 320 \)
Factor out \(2^x\):
\(2^x \left( \frac{1}{2} + 2 \right) = 320\)
Combine the terms inside the parentheses:
\( \frac{1}{2} + 2 = \frac{5}{2} \)
Now the equation is:
\(2^x \cdot \frac{5}{2} = 320\)
Multiply both sides by \(\frac{2}{5}\):
\(2^x = 320 \cdot \frac{2}{5} = 128\)
Since \(128 = 2^7\), we find:
\(x = 7\)
Thus, the correct answer is:
Final Answer: \(\boxed{7}\)
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