Solve: $36^{3s} = 216^{2s+1}$
The given equation is \(36^{3s} = 216^{2s+1}\). To solve for \(s\), we identify that both 36 and 216 can be expressed as powers of the same base, 6. Specifically, we have \(36 = 6^2\) and \(216 = 6^3\). By substituting these into the equation, we obtain:
\((6^2)^{3s} = (6^3)^{2s+1}\)
Using the power rule for exponents \((a^m)^n = a^{m \cdot n}\), we simplify the expressions on both sides:
On the left, \((6^2)^{3s} = 6^{2 \cdot 3s} = 6^{6s}\).
On the right, \((6^3)^{2s+1} = 6^{3 \cdot (2s+1)} = 6^{6s+3}\).
Equating the exponents, we have the linear equation \(6s = 6s + 3\).
Subtracting \(6s\) from both sides of the equation \(6s = 6s + 3\) results in \(0 = 3\). Since this is a false statement (a contradiction), there is no value of \(s\) that makes the equation true. Thus, there is no solution to the given equation.
Equations involving exponential functions are frequently used in modeling growth or decay in various fields such as finance, biology, or physics. For instance, if you were comparing two different investment strategies where one grows based on a doubling period and another based on a tripling period, you might find yourself setting up an equation to find the time when the investments become equal. Solving these helps determine if and when such a balance point actually occurs, which is crucial for making informed financial or scientific decisions.