Simplify: `(x^(a)/x^(b))^(a+b). (x^(b)/x^(c ))^(b+c) .(x^( c)/x^(a))^(c+a)`

Answer

To simplify the expression \((\frac{x^a}{x^b})^{a+b} \cdot (\frac{x^b}{x^c})^{b+c} \cdot (\frac{x^c}{x^a})^{c+a}\), we can follow these steps:

### Step 1: Apply the Quotient Rule
Using the property of exponents that states \(\frac{x^m}{x^n} = x^{m-n}\), we can rewrite each fraction in the expression:

\[ \frac{x^a}{x^b} = x^{a-b}, \quad \frac{x^b}{x^c} = x^{b-c}, \quad \frac{x^c}{x^a} = x^{c-a} \]

### Step 2: Rewrite the Expression
Now we can rewrite the entire expression using the results from Step 1:

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

### Step 3: Apply the Power of a Power Rule
Using the property \((x^m)^n = x^{m \cdot n}\), we can simplify each term:

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

### Step 4: Combine the Exponents
Since we are multiplying powers with the same base, we can add the exponents:

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

### Step 5: Expand Each Term
Now we will expand each term in the exponent:

1. \((a-b)(a+b) = a^2 - b^2\)
2. \((b-c)(b+c) = b^2 - c^2\)
3. \((c-a)(c+a) = c^2 - a^2\)

### Step 6: Combine the Expanded Terms
Now we combine the expanded terms:

\[ a^2 - b^2 + b^2 - c^2 + c^2 - a^2 \]

### Step 7: Simplify the Expression
Notice that the terms cancel out:

\[ a^2 - a^2 + b^2 - b^2 + c^2 - c^2 = 0 \]

### Final Result
Thus, the entire expression simplifies to:

\[ x^0 = 1 \]