$$ 3 ^ { 2 } \cdot 6 ^ { x - 1 } - 2 ^ { 3 } \cdot 6 ^ { x - 1 } = \frac { 1 } { 6 } $$

Question

UpStudy AI · Accepted Answer

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.
$$ x = 0 $$

\( 3 ^ { 2 } \cdot 6 ^ { x - 1 } - 2 ^ { 3 } \cdot 6 ^ { x - 1 } = \frac { 1 } { 6 } \)