Simplify :
`((2^(a))/(2^(b)))^(a+b).((2^(b))/(2^(c)))^(b+c).((2^(c))/(2^(a)))^(c+a)`

Answer

To simplify the expression

\[ \left(\frac{2^a}{2^b}\right)^{a+b} \cdot \left(\frac{2^b}{2^c}\right)^{b+c} \cdot \left(\frac{2^c}{2^a}\right)^{c+a} \]

we can follow these steps:

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

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

### Step 2: Substitute back into the expression
Now, substituting these back into the original expression, we have:

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

### Step 3: Apply the power of a power property
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify further:

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

### Step 4: Combine the exponents
Now we can combine these:

\[ 2^{(a-b)(a+b) + (b-c)(b+c) + (c-a)(c+a)} \]

### Step 5: Expand the products
Now we expand each term:

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

Combining these gives:

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

### Step 6: Simplify the expression
Notice that the terms cancel out:

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

### Step 7: Final expression
Thus, we have:

\[ 2^0 = 1 \]

### Final Answer
The simplified expression is:

\[ 1 \]