Select the correct answer.
Solve the exponential equation for $x$.
$256=\left(\frac{1}{4}\right)^{3 x+2}$
A. $x=2$
B. $x=-4$
C. $x=4$
D. $x=-2$
Answer (2)
Rewrite the equation with the same base: 4 4 = ( 4 − 1 ) 3 x + 2 .
Simplify the exponent: 4 4 = 4 − ( 3 x + 2 ) .
Equate the exponents: 4 = − ( 3 x + 2 ) .
Solve for x : x = − 2 .
The final answer is − 2 .
Explanation
Understanding the Problem We are given the exponential equation 256 = ( 4 1 ) 3 x + 2 . Our goal is to solve for x .
Rewriting with the Same Base First, we rewrite both sides of the equation with the same base. We know that 256 = 4 4 and 4 1 = 4 − 1 . Therefore, the equation becomes 4 4 = ( 4 − 1 ) 3 x + 2 .
Simplifying the Exponent Next, we simplify the right side of the equation using the power of a power rule, which states that ( a m ) n = a mn . So, ( 4 − 1 ) 3 x + 2 = 4 − ( 3 x + 2 ) . Now our equation is 4 4 = 4 − ( 3 x + 2 ) .
Equating the Exponents Since the bases are equal, we can set the exponents equal to each other: 4 = − ( 3 x + 2 ) .
Solving for x Now we solve the linear equation for x : 4 = − 3 x − 2 .
Isolating the x Term Add 2 to both sides of the equation: 4 + 2 = − 3 x − 2 + 2 , which simplifies to 6 = − 3 x .
Finding the Value of x Divide both sides by -3: − 3 6 = − 3 − 3 x , which simplifies to x = − 2 .
Final Answer Therefore, the solution to the exponential equation is x = − 2 . The correct answer is D.
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 a bank 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.
To solve the equation 256 = ( 4 1 ) 3 x + 2 , we rewrite both sides with a common base and solve for x . This leads us to find that the correct answer is x = − 2 . Therefore, the answer is option D.
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