Solve: `3^(4+1)-2xx3^(2x+2)-81=0`

Answer

To solve the equation \( 3^{4+1} - 2 \cdot 3^{2x+2} - 81 = 0 \), we can follow these steps:

### Step 1: Simplify the equation
First, we simplify \( 3^{4+1} \):
\[ 3^{4+1} = 3^5 \]
So the equation becomes:
\[ 3^5 - 2 \cdot 3^{2x+2} - 81 = 0 \]

### Step 2: Calculate \( 3^5 \) and \( 81 \)
Next, we calculate \( 3^5 \) and \( 81 \):
\[ 3^5 = 243 \quad \text{and} \quad 81 = 3^4 \]
Now, substituting these values back into the equation gives us:
\[ 243 - 2 \cdot 3^{2x+2} - 81 = 0 \]

### Step 3: Combine like terms
Now, we combine \( 243 \) and \( -81 \):
\[ 243 - 81 = 162 \]
So the equation simplifies to:
\[ 162 - 2 \cdot 3^{2x+2} = 0 \]

### Step 4: Isolate the term with \( x \)
Rearranging the equation gives:
\[ 2 \cdot 3^{2x+2} = 162 \]
Now, divide both sides by \( 2 \):
\[ 3^{2x+2} = \frac{162}{2} = 81 \]

### Step 5: Rewrite \( 81 \) as a power of \( 3 \)
We know that:
\[ 81 = 3^4 \]
So we can rewrite the equation as:
\[ 3^{2x+2} = 3^4 \]

### Step 6: Equate the exponents
Since the bases are the same, we can set the exponents equal to each other:
\[ 2x + 2 = 4 \]

### Step 7: Solve for \( x \)
Now, we solve for \( x \):
\[ 2x = 4 - 2 \]
\[ 2x = 2 \]
\[ x = 1 \]

### Final Answer
Thus, the solution to the equation is:
\[ \boxed{1} \]