## Simplifying Expressions with Exponents: (7xy^2z^5)^2

This article will delve into simplifying the expression **(7xy^2z^5)^2**. We'll break down the process step by step, applying the rules of exponents.

### Understanding the Concept

The expression **(7xy^2z^5)^2** represents the product of the base (7xy^2z^5) multiplied by itself twice.

### Applying the Rules of Exponents

**1. Distributing the exponent:**

The exponent 2 outside the parentheses applies to each term within the parentheses.

**(7xy^2z^5)^2 = 7^2 * x^2 * (y^2)^2 * (z^5)^2**

**2. Simplifying each term:**

- 7^2 = 49
- x^2 remains as x^2
- (y^2)^2 = y^(2*2) = y^4
- (z^5)^2 = z^(5*2) = z^10

**3. Combining the simplified terms:**

49 * x^2 * y^4 * z^10 = **49x^2y^4z^10**

### Final Result

Therefore, the simplified form of **(7xy^2z^5)^2** is **49x^2y^4z^10**.

### Key Takeaways

- When an exponent is applied to a product, it is distributed to each factor within the product.
- When an exponent is applied to another exponent, the exponents are multiplied together.

By understanding and applying these rules, we can effectively simplify expressions involving exponents.