To simplify the expression [tex]\(x^{(a-b)} \cdot x^{(b-c)} \cdot x^{(c-a)}\)[/tex], we need to use the properties of exponents.
### Step 1: Combine the exponents
Recall the property of exponents that states [tex]\(x^m \cdot x^n = x^{m+n}\)[/tex]. In other words, when you multiply expressions with the same base, you add the exponents.
Apply this property to our expression:
[tex]\[ x^{(a-b)} \cdot x^{(b-c)} \cdot x^{(c-a)} = x^{(a-b + b-c + c-a)} \][/tex]
### Step 2: Simplify the exponent
Now, we need to simplify the exponent [tex]\(a-b + b-c + c-a\)[/tex].
Combine the terms in the exponent:
[tex]\[ a - b + b - c + c - a \][/tex]
### Step 3: Cancel out terms
Observe that the terms [tex]\(b\)[/tex] and [tex]\(-b\)[/tex], [tex]\(c\)[/tex] and [tex]\(-c\)[/tex], [tex]\(a\)[/tex] and [tex]\(-a\)[/tex] cancel each other out:
[tex]\[ a - b + b - c + c - a = (a - a) + (b - b) + (c - c) = 0 \][/tex]
### Step 4: Resulting exponent
We now have:
[tex]\[ x^0 \][/tex]
### Step 5: Simplify using the zero exponent rule
Recall that any non-zero number raised to the power of 0 is 1:
[tex]\[ x^0 = 1 \][/tex]
### Final Answer
Thus, the simplified form of the expression [tex]\(x^{(a-b)} \cdot x^{(b-c)} \cdot x^{(c-a)}\)[/tex] is:
[tex]\[ 1 \][/tex]
