Question

If \(2^x + 2^{x+1} = 48\), then the value of \(x^x\) is

• A. 4
• B. 64
• C. 256
• D. 16

Hint:

Factor \(2^x\) out of the equation first.

Answer 1

The Correct Option is C

Step 1: Rewrite the equation to isolate \(2^x\) first.
$2^x + 2^{x+1} = 48$ can be seen as $2^x + 2 \cdot 2^x = 3 \cdot 2^x$, so $3 \cdot 2^x = 48$, which gives $2^x = 16$.

Step 2: Take $\log_2$ of both sides instead of matching powers directly.
$\log_2(2^x) = \log_2(16)$, so $x = \log_2(16)$.

Step 3: Evaluate the logarithm by counting powers of 2.
$2^1=2,\ 2^2=4,\ 2^3=8,\ 2^4=16$, so $\log_2(16) = 4$, meaning $x=4$.

Step 4: Compute $x^x$.
$x^x = 4^4 = 256$.

Final Answer:
$x^x = 256$. \[ \boxed{256} \]