Question
\( 3 ^ { 2 } \cdot 6 ^ { x - 1 } - 2 ^ { 3 } \cdot 6 ^ { x - 1 } = \frac { 1 } { 6 } \)
Ask by Allan Ayala. in South Africa
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = 0 \)
Solution
We start with the given equation:
3² · 6^(x - 1) - 2³ · 6^(x - 1) = 1/6
Step 1. Calculate the constants:
3² = 9 and 2³ = 8
So the equation becomes:
9 · 6^(x - 1) - 8 · 6^(x - 1) = 1/6
Step 2. Factor the common term 6^(x - 1):
(9 - 8) · 6^(x - 1) = 1/6
1 · 6^(x - 1) = 1/6
Thus:
6^(x - 1) = 1/6
Step 3. Write 1/6 as an exponent with base 6:
1/6 = 6^(-1)
So we have:
6^(x - 1) = 6^(-1)
Step 4. Equate the exponents (since the bases are the same):
x - 1 = -1
Step 5. Solve for x:
x = -1 + 1
x = 0
Therefore, the solution is x = 0.
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Mind Expander
Let’s first factor out \(6^{x-1}\) from the left side of the equation: \[ 6^{x-1} \cdot (3^2 - 2^3) = \frac{1}{6} \] Now we can simplify \(3^2 = 9\) and \(2^3 = 8\): \[ 6^{x-1} \cdot (9 - 8) = \frac{1}{6} \] This simplifies down to: \[ 6^{x-1} \cdot 1 = \frac{1}{6} \] Thus, we have: \[ 6^{x-1} = \frac{1}{6} \] Next, we can rewrite \(\frac{1}{6}\) as \(6^{-1}\): \[ 6^{x-1} = 6^{-1} \] Since the bases are the same, we can equate the exponents: \[ x - 1 = -1 \] Solving for \(x\): \[ x = 0 \] So, the solution is: \[ \boxed{0} \]
