Solve $6^{2 x+2} \cdot 6^{3 x}=1$.
$x=$
The given equation is \(6^{2x+2} \cdot 6^{3x} = 1\). To solve for \(x\), we first combine the terms on the left side using the exponent rule for products with the same base, which states that \(a^m \cdot a^n = a^{m+n}\). Applying this rule, we get:
\[6^{(2x+2) + 3x} = 1\]
\[6^{5x+2} = 1\]
We know that any non-zero number raised to the power of 0 is equal to 1. Specifically, since the base is 6, we can write 1 as \(6^0\). Thus, the equation becomes:
\[6^{5x+2} = 6^0\]
Since the bases are the same, we can equate the exponents:
\[5x + 2 = 0\]
Now we have a simple linear equation to solve for \(x\).
Subtracting 2 from both sides gives:
\[5x = -2\]
Dividing both sides by 5 yields:
\[x = -\frac{2}{5}\]
Thus, the solution to the equation is \(x = -\frac{2}{5}\), which can also be written as \(x = -0.4\).
This type of problem relates to finding time intervals in compound growth or decay scenarios. For instance, if a bacteria population grows according to an exponential function, solving for the exponent helps determine the specific time \(t\) when the population reaches a certain threshold. Understanding how to handle exponent rules is essential for modeling growth patterns in biology, finance, and engineering, where equations often involve unknown values within exponents.