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The expression to evaluate is:

\[
(x - y)(x + y) + (y - z)(y + z) + (z - x)(z + x)
\]

### Step 1: Expand each term using the difference of squares formula:

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

#### First term: \((x - y)(x + y)\)
\[
(x - y)(x + y) = x^2 - y^2
\]

#### Second term: \((y - z)(y + z)\)
\[
(y - z)(y + z) = y^2 - z^2
\]

#### Third term: \((z - x)(z + x)\)
\[
(z - x)(z + x) = z^2 - x^2
\]

### Step 2: Combine all terms:
\[
x^2 - y^2 + y^2 - z^2 + z^2 - x^2
\]

### Step 3: Simplify:
\[
x^2 - x^2 + y^2 - y^2 + z^2 - z^2 = 0
\]

Thus, the value of the entire expression is:

\[
\boxed{0}
\]

This matches the given answer.

Would you like further details or have any questions? Here are some related questions to explore this further:

1. What is the difference of squares formula and how is it used in expanding expressions?
2. Can similar expressions with more variables also simplify to zero?
3. How does simplifying expressions with multiple variables work in more complex problems?
4. What are some other ways to factor quadratic expressions?
5. How can we use algebraic identities to simplify polynomial expressions?

**Tip:** Always check if expressions can be factored or simplified using known algebraic identities before performing lengthy expansions.

The value of (x - y)(x + y) + (y - z)(y + z) + (z - x)(z + x) is:

Solve the expression (x - y)(x + y) + (y - z)(y + z) + (z - x)(z + x) using the difference of squares formula. Step-by-step explanation, suitable for high school students. Ideal for Grades 9-10.

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