Simplify :
`((x^(a))/(x^(-b)))^(a^(2)-ab+b^(2))xx((x^(b))/(x^(-c)))^(b^(2)-bc+c^(2))xx((x^(c))/(x^(-a)))^(c^(2)-ca+a^(2))`

Answer

To simplify the expression

\[ \left(\frac{x^a}{x^{-b}}\right)^{a^2 - ab + b^2} \cdot \left(\frac{x^b}{x^{-c}}\right)^{b^2 - bc + c^2} \cdot \left(\frac{x^c}{x^{-a}}\right)^{c^2 - ca + a^2} \]

we will follow the laws of exponents step by step.

### Step 1: Rewrite the fractions using the exponent rules

Using the property \(\frac{x^m}{x^n} = x^{m-n}\), we can rewrite each fraction:

\[ \frac{x^a}{x^{-b}} = x^{a - (-b)} = x^{a + b} \]
\[ \frac{x^b}{x^{-c}} = x^{b - (-c)} = x^{b + c} \]
\[ \frac{x^c}{x^{-a}} = x^{c - (-a)} = x^{c + a} \]

### Step 2: Substitute back into the expression

Now substituting these back into the original expression gives us:

\[ \left(x^{a + b}\right)^{a^2 - ab + b^2} \cdot \left(x^{b + c}\right)^{b^2 - bc + c^2} \cdot \left(x^{c + a}\right)^{c^2 - ca + a^2} \]

### Step 3: Apply the power of a power rule

Using the property \((x^m)^n = x^{m \cdot n}\), we can simplify each term:

\[ x^{(a + b)(a^2 - ab + b^2)} \cdot x^{(b + c)(b^2 - bc + c^2)} \cdot x^{(c + a)(c^2 - ca + a^2)} \]

### Step 4: Combine the exponents

Now we can combine the exponents since they are all multiplied:

\[ x^{(a + b)(a^2 - ab + b^2) + (b + c)(b^2 - bc + c^2) + (c + a)(c^2 - ca + a^2)} \]

### Step 5: Expand and simplify the exponent

Now we need to expand each term in the exponent:

1. **First term**:
\[ (a + b)(a^2 - ab + b^2) = a^3 + b^3 + a^2b - ab^2 \]

2. **Second term**:
\[ (b + c)(b^2 - bc + c^2) = b^3 + c^3 + b^2c - bc^2 \]

3. **Third term**:
\[ (c + a)(c^2 - ca + a^2) = c^3 + a^3 + c^2a - ca^2 \]

Now, combine all these terms:

\[ x^{(a^3 + b^3 + a^2b - ab^2) + (b^3 + c^3 + b^2c - bc^2) + (c^3 + a^3 + c^2a - ca^2)} \]

### Step 6: Combine like terms

Combining like terms, we get:

\[ x^{2a^3 + 2b^3 + 2c^3 + (a^2b + b^2c + c^2a) - (ab^2 + bc^2 + ca^2)} \]

### Final Result

Thus, the simplified expression is:

\[ x^{2(a^3 + b^3 + c^3) + (a^2b + b^2c + c^2a) - (ab^2 + bc^2 + ca^2)} \]