The value of `((x^(a))/(x^(b)))^(a+b) xx ((x^(b))/(x^(c)))^(b+c) xx ((x^(c))/(x^(a)))^(c + a)` is equal to

Options

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

Answer

To solve the expression

\[ \left(\frac{x^a}{x^b}\right)^{a+b} \cdot \left(\frac{x^b}{x^c}\right)^{b+c} \cdot \left(\frac{x^c}{x^a}\right)^{c+a} \]

we will follow these steps:

### Step 1: Simplify each term using the property of exponents
Using the property \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify each fraction:

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

So the expression becomes:

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

### Step 2: Apply the power of a power property
Using the property \((a^m)^n = a^{m \cdot n}\), we can further simplify:

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

Thus, the expression now looks like:

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

### Step 3: Combine the exponents
Using the property \(a^m \cdot a^n = a^{m+n}\), we can combine the exponents:

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

### Step 4: Expand the terms in the exponent
Now we will expand each term:

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

So the exponent becomes:

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

### Step 5: Simplify the exponent
Notice that when we add these terms together, they cancel out:

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

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

\[ x^0 = 1 \]

So the value of the original expression is:

\[ \boxed{1} \]