Solve the exponential equation for $x$.
$256=\left(\frac{1}{4}\right)^{3 x+2}$
A. $x=4$
B. $x=-2$
C. $x=2$
D. $x=-4$
Answer (2)
To solve the equation 256 = ( 4 1 ) 3 x + 2 , we rewrite both sides with base 4. This leads us to set the exponents equal, resulting in the solution x = − 2 . The correct answer is option B: x = − 2 .
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Rewrite both sides of the equation with the same base: 4 4 = ( 4 − 1 ) 3 x + 2 .
Simplify the right side using the power of a power rule: 4 4 = 4 − 3 x − 2 .
Set the exponents equal to each other: 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 and asked to solve for x .
Rewriting with the Same Base To solve this equation, we need to rewrite both sides with the same base. We know that 256 = 4 4 and 4 1 = 4 − 1 . Substituting these into the original equation, we get: 4 4 = ( 4 − 1 ) 3 x + 2
Simplifying the Equation Now, we simplify the right side of the equation using the power of a power rule, which states that ( a m ) n = a mn . Applying this rule, we have: 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 . First, add 2 to both sides: 4 + 2 = − 3 x − 2 + 2 6 = − 3 x Then, divide both sides by -3: − 3 6 = − 3 − 3 x 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 that earns compound interest, the amount of money you have after a certain period can be calculated using an exponential equation. Similarly, exponential equations can be used to determine the remaining amount of a radioactive substance after a certain period.
