Question
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = 3 \)
Solution
To solve the equation:
\[
2^{x} + 2^{x+1} = 24
\]
Let's break it down step by step.
1. **Simplify \(2^{x+1}\):**
Recall that \(2^{x+1} = 2 \times 2^{x}\).
\[
2^{x} + 2 \times 2^{x} = 24
\]
2. **Factor out \(2^{x}\):**
\[
2^{x} (1 + 2) = 24
\]
\[
2^{x} \times 3 = 24
\]
3. **Isolate \(2^{x}\):**
\[
2^{x} = \frac{24}{3}
\]
\[
2^{x} = 8
\]
4. **Express 8 as a power of 2:**
\[
8 = 2^{3}
\]
So,
\[
2^{x} = 2^{3}
\]
5. **Set the exponents equal to each other:**
\[
x = 3
\]
**Final Answer:**
\[
x = 3
\]
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Simplify this solution Beyond the Answer
Let's simplify the given equation step by step. First, notice that \( 2^{x+1} \) can be rewritten as \( 2 \cdot 2^{x} \). Hence, the equation transforms into: \[ 2^{x} + 2 \cdot 2^{x} = 24 \] This further simplifies to: \[ 2^{x} + 2^{x+1} = 2^{x} + 2^{x} \cdot 2 = 2^{x} (1 + 2) = 2^{x} \cdot 3 \] So, we have: \[ 3 \cdot 2^{x} = 24 \] Next, we solve for \( 2^{x} \) by dividing both sides by 3: \[ 2^{x} = \frac{24}{3} = 8 \] Now, since \( 8 \) can be expressed as a power of \( 2 \) (specifically \( 2^3 \)), we set: \[ 2^{x} = 2^{3} \] From this, we can equate the exponents: \[ x = 3 \] Thus, the solution for \( x \) is: \[ \boxed{3} \]
