Simplify :
`((x^a)/(x^b))^(4)xx((x^b)/(x^c))^(3)xx((x^c)/(x^a))^2`

Options

A. `0`
B. `x`
C. `1`
D. `x^(2a-b-c)`

Answer

To simplify the expression \(\left(\frac{x^a}{x^b}\right)^4 \cdot \left(\frac{x^b}{x^c}\right)^3 \cdot \left(\frac{x^c}{x^a}\right)^2\), we can follow these steps:

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

\[ \left(\frac{x^a}{x^b}\right)^4 = (x^{a-b})^4 \]
\[ \left(\frac{x^b}{x^c}\right)^3 = (x^{b-c})^3 \]
\[ \left(\frac{x^c}{x^a}\right)^2 = (x^{c-a})^2 \]

### Step 2: Apply the Power of a Power Rule
Next, we apply the power of a power rule, which states \((x^m)^n = x^{m \cdot n}\):

\[ (x^{a-b})^4 = x^{4(a-b)} = x^{4a - 4b} \]
\[ (x^{b-c})^3 = x^{3(b-c)} = x^{3b - 3c} \]
\[ (x^{c-a})^2 = x^{2(c-a)} = x^{2c - 2a} \]

### Step 3: Combine the Exponents
Now we can combine these expressions since they all have the same base \(x\):

\[ x^{4a - 4b} \cdot x^{3b - 3c} \cdot x^{2c - 2a} = x^{(4a - 4b) + (3b - 3c) + (2c - 2a)} \]

### Step 4: Simplify the Exponent
Now, we simplify the exponent:

\[ = x^{4a - 4b + 3b - 3c + 2c - 2a} \]

Combining like terms:

\[ = x^{(4a - 2a) + (-4b + 3b) + (-3c + 2c)} \]
\[ = x^{2a - b - c} \]

### Final Answer
Thus, the simplified expression is:

\[ \boxed{x^{2a - b - c}} \]