We are given the equation:

$4^{(x+1)} = 2$

Step 1: Express $4$ as $2^2$

We know that $4 = 2^2$, so we can rewrite the equation as:

$(2^2)^{(x+1)} = 2$

Step 2: Apply the power of a power property

Using the property of exponents $(a^m)^n = a^{m \cdot n}$, we can simplify the left-hand side:

$2^{2(x+1)} = 2$

Step 3: Simplify the exponent

Now expand the exponent on the left-hand side:

$2^{2x + 2} = 2^1$

Step 4: Set the exponents equal to each other

Since the bases are the same (both are powers of 2), we can set the exponents equal to each other:

$2x + 2 = 1$

Step 5: Solve for $x$

Now solve for $x$:

$2x = 1 - 2$ $2x = -1$ $x = -\frac{1}{2}$

Step 6: Find $4^{(x-1)}$

We are asked to express $4^{(x-1)}$ as a common fraction. First, substitute $x = -\frac{1}{2}$ into $4^{(x-1)}$:

$4^{(x-1)} = 4^{\left(-\frac{1}{2} - 1\right)} = 4^{-\frac{3}{2}}$

Step 7: Simplify the expression

Now express $4$ as $2^2$:

$4^{-\frac{3}{2}} = (2^2)^{-\frac{3}{2}} = 2^{-3}$

Since $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$, we have:

$4^{(x-1)} = \frac{1}{8}$

Final Answer:

$4^{(x-1)} = \frac{1}{8}$

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Would you like further clarification on any step? Here are 5 follow-up questions to consider:

1. What is the general method for solving exponential equations?
2. How do you apply the power of a power rule for exponents?
3. Why do we set the exponents equal to each other when bases are the same?
4. How do you convert negative exponents to fractions?
5. What other techniques can be used to solve equations involving exponents?

Tip: When dealing with equations involving exponents with the same base, always consider simplifying the base first to avoid more complicated expressions.