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The distributive property is a fundamental principle in algebra that helps us simplify expressions, especially when dealing with multiplication over addition or subtraction inside parentheses. It's like spreading a single term outside the parentheses across all terms inside them. For instance, if you have an expression like \((a + b)c\), using the distributive property means you will multiply \(c\) by both \(a\) and \(b\), resulting in \(ac + bc\).
In the exercise \((x^4 + x^2)x^3\), the term \(x^3\) outside the parentheses needs to be multiplied by each term inside the parentheses, which is what we mean by applying the distributive property. This results in two separate products: \(x^3 \cdot x^4\) and \(x^3 \cdot x^2\). These calculations set the groundwork for using exponent rules effectively in the next steps.

Exponent rules are crucial for simplifying expressions involving powers of the same base. When you multiply like bases, you add the exponents according to the rule:

• \(a^m \cdot a^n = a^{m+n}\)

In our exercise, we apply this rule as follows:
- For the first term, \(x^3 \cdot x^4\), you add the exponents 3 and 4, which gives you \(x^{7}\).
- For the second term, \(x^3 \cdot x^2\), adding the exponents 3 and 2 gives you \(x^{5}\).
These calculations are straightforward once you remember the exponent rule for multiplying powers of the same base. The key is to focus on combining like bases by simply adding their exponents.

Simplifying expressions involves combining like terms and reducing the expression to its simplest form. In our case, once we've applied the exponent rules, we are left with \(x^7 + x^5\).
Each term in this expression is already in its simplest form since no like terms are present to combine further (i.e., they each have different exponents). Therefore, the final expression, \(x^{7} + x^{5}\), is indeed the simplest form of the original exercise.
Simplifying ensures that the expression is concise and easier to interpret, which is especially helpful for further mathematical operations.