Question
Solve the following for \( x \) 1. \( 2^{x+1}-2^{x}=8 \)
Ask by Bond Wheeler. in South Africa
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = 3 \)
Solution
To solve the equation \( 2^{x+1} - 2^{x} = 8 \) for \( x \), follow these steps:
1. **Simplify the Left Side:**
Recall that \( 2^{x+1} = 2 \cdot 2^{x} \). Substitute this into the equation:
\[
2 \cdot 2^{x} - 2^{x} = 8
\]
2. **Factor Out \( 2^{x} \):**
Factor \( 2^{x} \) from the left side:
\[
(2 - 1) \cdot 2^{x} = 8
\]
Simplify:
\[
1 \cdot 2^{x} = 8 \quad \Rightarrow \quad 2^{x} = 8
\]
3. **Solve for \( x \):**
Recognize that \( 8 \) is a power of \( 2 \):
\[
8 = 2^{3}
\]
Therefore:
\[
2^{x} = 2^{3} \quad \Rightarrow \quad x = 3
\]
**Final Answer:** \( x = 3 \)
Reviewed and approved by the UpStudy tutoring team
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Beyond the Answer
To solve the equation \( 2^{x+1} - 2^x = 8 \), we can first rewrite \( 2^{x+1} \) as \( 2 \cdot 2^x \): \[ 2 \cdot 2^x - 2^x = 8 \] This simplifies to: \[ (2 - 1) \cdot 2^x = 8 \] Which further simplifies to: \[ 2^x = 8 \] Since \( 8 \) can be rewritten as \( 2^3 \), we have: \[ 2^x = 2^3 \] Thus, we can equate the exponents: \[ x = 3 \] Therefore, the solution is \[ \boxed{3} \]