Solve the exponential equation for $x$.
$216=6^{(4 x+11)}$
We are given the exponential equation \(216 = 6^{(4x + 11)}\). To solve for \(x\), we first look to express both sides of the equation with the same base. Recognizing that \(216\) is a power of \(6\), we can calculate:
\(6^1 = 6\)
\(6^2 = 36\)
\(6^3 = 36 \times 6 = 216\)
Thus, we can rewrite the left side of the equation as \(6^3\).
Now that we have the equation in the form \(6^3 = 6^{(4x + 11)}\), we can use the property of exponential functions. Since the base is the same and the exponential function \(f(y) = 6^y\) is one-to-one, we can equate the exponents:
\(3 = 4x + 11\)
We now have a simple linear equation to solve for \(x\):
\(3 = 4x + 11\)
Subtract \(11\) from both sides:
\(3 - 11 = 4x\)
\(-8 = 4x\)
Divide by \(4\):
\(x = \frac{-8}{4}\)
\(x = -2\)
The calculated value of \(x\) is \(-2\). Checking the given choices (\(x = -3\), \(x = 2\), \(x = 3\), \(x = -2\)), we confirm that the correct answer is \(x = -2\). Thus, the solution is \(\boxed{-2}\).
Exponential equations are frequently used in fields like finance and biology to model growth or decay. For example, if a bacteria culture triples every hour, we might model its population using an exponential function. Similarly, in compound interest, the value of an investment over time is calculated using an exponential formula. Solving these equations helps us predict future values or determine how long it takes for a system to reach a certain size.