Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x=2 \)
Solution
Solve the equation \( 3^{x+1}+3^{x-1}+3^{x}=39 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(3^{x+1}+3^{x-1}+3^{x}=39\)
- step1: Add the terms:
\(13\times 3^{x-1}=39\)
- step2: Divide both sides:
\(\frac{13\times 3^{x-1}}{13}=\frac{39}{13}\)
- step3: Divide the numbers:
\(3^{x-1}=3\)
- step4: Set the exponents equal:
\(x-1=1\)
- step5: Move the constant to the right side:
\(x=1+1\)
- step6: Add the numbers:
\(x=2\)
The solution to the equation \(3^{x+1}+3^{x-1}+3^{x}=39\) is \(x=2\).
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Simplify this solution Beyond the Answer
To tackle the equation \( 3^{x+1} + 3^{x-1} + 3^{x} = 39 \), we can simplify it by factoring out \( 3^{x-1} \). This gives us: \[ 3^{x+1} = 3 \cdot 3^{x} \quad \text{and} \quad 3^{x-1} = \frac{3^{x}}{3} \] Rewriting the equation, we have: \[ 3 \cdot 3^{x} + \frac{3^{x}}{3} + 3^{x} = 39 \] Now, if we denote \( 3^{x} \) as \( y \), the equation transforms into: \[ 3y + \frac{y}{3} + y = 39 \] Multiplying the entire equation by 3 to eliminate the fraction: \[ 9y + y + 3y = 117 \] This simplifies to: \[ 13y = 117 \] Thus, \( y = 9 \), which means \( 3^x = 9 \). Therefore, we find \( x = 2 \). So, the solution is \( x = 2 \).
