Solve for x: 4^(3-x) = 1/64
Understand the Problem
The question asks us to solve the exponential equation for the variable x. We will first express both sides of the equation with the same base, then equate the exponents to solve for x.
Answer
$x=6$
Answer for screen readers
$x = 6$
Steps to Solve
- Express both sides with the same base
We want to express both sides of the equation $4^{3-x} = \frac{1}{64}$ with base 4 First, rewrite $\frac{1}{64}$ as a power of 4. Since $64 = 4^3$, we can write $\frac{1}{64} = \frac{1}{4^3} = 4^{-3}$. Thus, the equation becomes $4^{3-x} = 4^{-3}$.
- Equate the exponents
Since the bases are equal, we can set the exponents equal to each other: $3-x = -3$
- Solve for x
Add $x$ to both sides of the equation: $3-x+x = -3+x$ $3 = -3 + x$ Add 3 to both sides: $3 + 3 = -3 + 3 + x$ $6 = x$ Therefore, $x = 6$.
$x = 6$
More Information
We have successfully solved the exponential equation by expressing both sides with the same base and solving for $x$.
Tips
A common mistake is not being able to express both sides of the equation with the same base. Another common mistake is incorrectly manipulating the equation when solving for $x$. Always double-check your work to ensure each step is correct.
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