Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x=1 \)
Solution
Solve the equation \( 3^{x+1}=8+3^{x-1} \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(3^{x+1}=8+3^{x-1}\)
- step1: Move the expression to the left side:
\(3^{x+1}-3^{x-1}=8\)
- step2: Calculate:
\(8\times 3^{x-1}=8\)
- step3: Divide both sides:
\(\frac{8\times 3^{x-1}}{8}=\frac{8}{8}\)
- step4: Divide the numbers:
\(3^{x-1}=1\)
- step5: Rewrite in exponential form:
\(3^{x-1}=3^{0}\)
- step6: Set the exponents equal:
\(x-1=0\)
- step7: Move the constant to the right side:
\(x=0+1\)
- step8: Remove 0:
\(x=1\)
The solution to the equation \(3^{x+1}=8+3^{x-1}\) is \(x=1\).
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Simplify this solution The Deep Dive
To solve the equation \( 3^{x+1} = 8 + 3^{x-1} \), we can start by rewriting \( 3^{x+1} \) as \( 3 \cdot 3^x \) and \( 3^{x-1} \) as \( \frac{3^x}{3} \): \[ 3 \cdot 3^x = 8 + \frac{3^x}{3} \] Now, let's multiply the entire equation by 3 to eliminate the fraction: \[ 9 \cdot 3^x = 24 + 3^x \] Next, we can rearrange the equation by moving \( 3^x \) to the left side: \[ 9 \cdot 3^x - 3^x = 24 \] This simplifies to: \[ 8 \cdot 3^x = 24 \] Now, divide both sides by 8: \[ 3^x = 3 \] Taking the logarithm base 3 of both sides, or recognizing that \( 3^1 = 3 \), we find: \[ x = 1 \] Thus, the solution to the equation is: \[ \boxed{1} \]
