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

To find the value of \(x^x\) given the equation \(2^x + 2^{x+1} = 48\), we start by simplifying the equation.

The term \(2^{x+1}\) can be rewritten using the property of exponents as \(2 \times 2^x\). Therefore, the equation becomes:

\(2^x + 2 \times 2^x = 48\)

Combining like terms, we factor out \(2^x\): 

\(2^x (1 + 2) = 48\)

This simplifies to:

\(2^x \times 3 = 48\)

Solving for \(2^x\), we divide both sides by 3:

\(2^x = \frac{48}{3} = 16\)

We know \(16\) is \(2^4\), so:

\(2^x = 2^4\)

By comparing the exponents, we find:

\(x = 4\)

Now, we calculate \(x^x\) using this value of \(x\):

\(x^x = 4^4\)

Calculating \(4^4\):

\(4^4 = 4 \times 4 \times 4 \times 4 = 256\)

Therefore, the value of \(x^x\) is 256.

Thus, the correct option is 256.

Answer 2

Step 1: Write \(2^{x+1}\) in terms of \(2^x\).
\(2^{x+1} = 2^x \times 2\), so the equation \(2^x + 2^{x+1} = 48\) becomes \(2^x + 2 \times 2^x = 48\).

Step 2: Combine like terms and solve for \(2^x\).
\(2^x(1+2) = 48\), so \(3 \times 2^x = 48\), giving \(2^x = 16\).

Step 3: Solve for x.
Since \(16 = 2^4\), we get \(2^x = 2^4\), so \(x = 4\).

Step 4: Find \(x^x\).
\(x^x = 4^4 = 4 \times 4 \times 4 \times 4 = 256\).

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