Solve : `root(4)(|x-3|^(x+1))=root(3)(|x-3|^(x-2))`.

Explanation

To solve the equation \( \sqrt[4]{|x-3|^{(x+1)}} = \sqrt[3]{|x-3|^{(x-2)}} \), we can follow these steps:

### Step 1: Rewrite the equation in exponential form
We can rewrite the roots in terms of exponents:
\[ |x-3|^{(x+1)/4} = |x-3|^{(x-2)/3} \]

### Step 2: Set the exponents equal to each other
Since the bases are the same (assuming \( |x-3| \neq 0 \)), we can set the exponents equal to each other:
\[ \frac{x+1}{4} = \frac{x-2}{3} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ 3(x+1) = 4(x-2) \]

### Step 4: Expand both sides
Expanding both sides results in:
\[ 3x + 3 = 4x - 8 \]

### Step 5: Rearrange the equation
Rearranging the equation to isolate \( x \):
\[ 3 + 8 = 4x - 3x \]
\[ 11 = x \]

### Step 6: Solve for \( |x-3| = 1 \)
Now, we also need to consider the case when \( |x-3| = 1 \):
1. \( x - 3 = 1 \) gives \( x = 4 \)
2. \( x - 3 = -1 \) gives \( x = 2 \)

### Step 7: List all solutions
The solutions we found are:
1. \( x = 11 \)
2. \( x = 4 \)
3. \( x = 2 \)

### Final Solutions
Thus, the complete solution set is:
\[ x = 2, \quad x = 4, \quad x = 11 \]

---