Let's break down the given expression step by step:

Expression:  
\[
(xyz)^3 + x^2y + (xy)^0 + (xy^{12})^2
\]

### Step 1: Simplify each term
1. \((xyz)^3\): Apply the power rule \((ab)^n = a^n b^n\):
   \[
   (xyz)^3 = x^3y^3z^3
   \]

2. \(x^2y\): This term is already simplified.

3. \((xy)^0\): Anything raised to the power of 0 is 1:
   \[
   (xy)^0 = 1
   \]

4. \((xy^{12})^2\): Apply the power rule \((a^m)^n = a^{m \cdot n}\):
   \[
   (xy^{12})^2 = x^2(y^{12})^2 = x^2y^{24}
   \]

### Step 2: Substitute the simplified terms back into the expression:
\[
x^3y^3z^3 + x^2y + 1 + x^2y^{24}
\]

This is the simplified form of the given expression.

Would you like more details on any step or have additional questions?

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### Related Questions:
1. What is the power rule for exponents and how does it apply in algebra?
2. How does the distributive property work with powers in multiple variables?
3. Can you explain how to simplify expressions with exponents and variables further?
4. What happens when an expression contains negative exponents or fractional exponents?
5. How do you approach factoring expressions like these?

**Tip:** Always remember that any non-zero number raised to the power of 0 is 1!

Learn how to simplify the expression (xyz)^3 + x^2y + (xy)^0 + (xy^{12})^2 step by step. This tutorial covers exponentiation, the power rule, and the zero exponent rule, suitable for students in Grades 9-12.

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