Solve the following :
`(x^b/x^c)^a xx (x^c/x^a)^b xx (x^a/x^b)^c` is equal to :

Options

A. 0
B. 1
C. abc
D. none of these

Answer

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

### Step 1: Simplify Each Fraction
We start by simplifying each fraction in the expression.

1. \(\frac{x^b}{x^c} = x^{b-c}\)
2. \(\frac{x^c}{x^a} = x^{c-a}\)
3. \(\frac{x^a}{x^b} = x^{a-b}\)

### Step 2: Apply the Exponents
Now we will apply the exponents to each simplified fraction:

1. \((\frac{x^b}{x^c})^a = (x^{b-c})^a = x^{a(b-c)}\)
2. \((\frac{x^c}{x^a})^b = (x^{c-a})^b = x^{b(c-a)}\)
3. \((\frac{x^a}{x^b})^c = (x^{a-b})^c = x^{c(a-b)}\)

### Step 3: Combine the Exponents
Next, we will combine all the terms:

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

Using the property of exponents that states \(x^m \cdot x^n = x^{m+n}\), we can combine the exponents:

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

### Step 4: Expand the Exponents
Now we will expand the expression in the exponent:

1. \(a(b-c) = ab - ac\)
2. \(b(c-a) = bc - ab\)
3. \(c(a-b) = ac - bc\)

Now we add these together:

\[ ab - ac + bc - ab + ac - bc \]

### Step 5: Simplify the Expression
Notice that the terms \(ab\), \(-ab\), \(-ac\), and \(ac\) cancel out, as do \(bc\) and \(-bc\):

\[ ab - ab - ac + ac + bc - bc = 0 \]

### Step 6: Final Result
Thus, we have:

\[ x^0 \]

And we know that any non-zero number raised to the power of 0 is equal to 1:

\[ x^0 = 1 \]

Therefore, the final answer is:

\[ \boxed{1} \]