Solve the exponential equation for $x$.
$216=6^{(4 x+11)}$
A. $x=2$
B. $x=-2$
C. $x=3$
D. $x=-3
Answer (2)
Rewrite 216 as a power of 6: 216 = 6 3 .
Substitute into the equation: 6 3 = 6 ( 4 x + 11 ) .
Equate the exponents: 3 = 4 x + 11 .
Solve for x : x = − 2 .
Explanation
Understanding the Problem We are given the exponential equation 216 = 6 ( 4 x + 11 ) . Our goal is to solve for x .
Rewriting the Equation First, we need to express 216 as a power of 6. We know that 6 1 = 6 , 6 2 = 36 , and 6 3 = 216 . So, we can rewrite the equation as: 6 3 = 6 ( 4 x + 11 )
Equating the Exponents Since the bases are equal, we can equate the exponents: 3 = 4 x + 11
Solving for x Now, we solve the linear equation for x . Subtract 11 from both sides: 3 − 11 = 4 x + 11 − 11 − 8 = 4 x
Finding the Value of x Divide both sides by 4: 4 − 8 = 4 4 x − 2 = x So, x = − 2 .
Final Answer Therefore, the solution to the exponential equation is x = − 2 .
Examples
Exponential equations are used in various real-world applications, such as modeling population growth, radioactive decay, and compound interest. For example, if you invest money in an account with compound interest, the amount of money you have after a certain time can be modeled using an exponential equation. Solving these equations helps you determine how long it will take for your investment to reach a certain value. Understanding exponential equations is crucial in finance, science, and engineering.
The solution to the exponential equation 216 = 6 ( 4 x + 11 ) is x = − 2 , which corresponds to option B. We derived this by rewriting 216 as a power of 6, equating the exponents, and solving for x. Therefore, the correct answer is B. x = -2.
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