Solve:
$64^{-3 x-3} \cdot 64+22=38$
A. $x=-\frac{9}{8}$
B. $x=-\frac{8}{9}$
C. $x=\frac{8}{9}$
D. $x=\frac{9}{8}$
Answer (1)
Subtract 22 from both sides: 6 4 − 3 x − 3 "."64 = 16 .
Simplify using exponent rules: 6 4 − 3 x − 2 = 16 .
Rewrite with a common base: 4 − 9 x − 6 = 4 2 .
Solve for x: x = − 9 8 .
x = − 9 8
Explanation
Understanding the Problem We are given the equation 6 4 − 3 x − 3 "."64 + 22 = 38 and asked to solve for x . Let's break down the steps to isolate x .
Isolating the Term with x First, we subtract 22 from both sides of the equation to isolate the term with the variable: 6 4 − 3 x − 3 "."64 = 38 − 22
Simplifying the Equation Simplifying the right side, we get: 6 4 − 3 x − 3 "."64 = 16
Rewriting 64 We can rewrite 64 as 6 4 1 , so the equation becomes: 6 4 − 3 x − 3 "."6 4 1 = 16
Using Exponent Properties Using the property of exponents a m "." a n = a m + n , we simplify the left side: 6 4 − 3 x − 3 + 1 = 16
6 4 − 3 x − 2 = 16
Changing the Base Now, we rewrite both sides with the same base. Since 64 = 4 3 and 16 = 4 2 , the equation becomes: ( 4 3 ) − 3 x − 2 = 4 2
Simplifying Exponents Using the property ( a m ) n = a mn , we simplify the left side: 4 3 ( − 3 x − 2 ) = 4 2
4 − 9 x − 6 = 4 2
Equating Exponents Since the bases are equal, the exponents must be equal: − 9 x − 6 = 2
Solving for x Now, we solve for x . Add 6 to both sides: − 9 x = 2 + 6
− 9 x = 8
Finding the Value of x Divide both sides by -9: x = − 9 8
x = − 9 8
Final Answer Therefore, the solution to the equation is x = − 9 8 .
Examples
Imagine you are designing a communication system where signals decay exponentially with distance. Solving equations like this helps determine the optimal signal strength needed to reach a specific receiver, ensuring reliable communication. Understanding exponential decay and solving for variables is crucial in fields like telecommunications, acoustics, and even financial modeling.
