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Understanding the distributive law is essential for simplifying algebraic expressions. This fundamental property allows you to multiply a single term by each term within a parenthesis. The distributive law asserts that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. In mathematical terms, for any real numbers a, b, and c, the distributive law is written as \(a(b + c) = ab + ac\).

In practice, if you encounter an expression like \(2(5x + 4)\), you distribute the 2 to both \(5x\) and \(4\), resulting in \(2 \times 5x + 2 \times 4\). This step transforms the given expression by removing the parentheses and setting the stage for further simplification.

To streamline an algebraic expression, it's crucial to combine like terms. Like terms are terms that contain the same variables raised to the same power. The coefficients of these terms can be different. To combine like terms, you add or subtract the coefficients while keeping the variables unchanged.

For example, after applying the distributive law to \(10x + 8 - 3\), you will notice that \(8\) and \( -3\) are like terms because they are both constants (terms without a variable). When you combine these like terms by performing the operation \(8 - 3\), you end up with \(5\). The expression now reads \(10x - 5\), with no further like terms to combine. Remember, variables must be identical for terms to be combined, so in this case, \(10x\) cannot be combined with any other term.

Multiplication is a basic arithmetic operation that is frequently performed to simplify algebraic expressions. When you multiply terms in an algebraic expression, you multiply their coefficients and keep the variables the same. After applying the distributive law, multiplication is usually the next step.

In the expression \(2 \times 5x + 2 \times 4 - 3\), the multiplications to be performed are \(2 \times 5x\) and \(2 \times 4\). The first yields \(10x\) and the second yields \(8\), leading to the new expression \(10x + 8 - 3\). It is only after performing these multiplications that we can proceed to combine like terms, as demonstrated in the previous steps. This process is crucial for simplifying an expression to its most reduced form.