To solve the equation \( 9^{x+1} = 81 \), we can follow these steps:

### Step 1: Express the numbers as powers of the same base.
We know that both 9 and 81 can be expressed as powers of 3:
\[
9 = 3^2 \quad \text{and} \quad 81 = 3^4
\]
So, the equation becomes:
\[
(3^2)^{x+1} = 3^4
\]

### Step 2: Simplify the equation using the power rule.
Using the power rule \((a^m)^n = a^{m \cdot n}\), we can simplify the left side:
\[
3^{2(x+1)} = 3^4
\]

### Step 3: Set the exponents equal to each other.
Since the bases are the same, the exponents must be equal:
\[
2(x + 1) = 4
\]

### Step 4: Solve for \(x\).
Distribute the 2 on the left side:
\[
2x + 2 = 4
\]
Subtract 2 from both sides:
\[
2x = 2
\]
Divide both sides by 2:
\[
x = 1
\]

### Solution:
The value of \(x\) is \( \boxed{1} \).

Would you like any further details or have any questions?

### Related Questions:
1. How would the solution change if the equation was \(9^{x+1} = 243\)?
2. What is the general process for solving exponential equations?
3. How do you solve \(9^{2x} = 81\)?
4. How can logarithms be used to solve exponential equations like this?
5. How do you simplify \( (a^m)^n \) in different scenarios?

### Tip:
Always try to express numbers with a common base when solving exponential equations. This simplifies the process and makes it easier to

Learn how to solve the exponential equation 9^(x+1) = 81 step-by-step. This problem involves using the power rule and simplifying exponents. Suitable for high school students studying exponential equations.

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