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Understanding the distributive property is crucial when simplifying algebraic expressions. When you encounter a multiplication that involves a sum or difference inside parentheses, the distributive property allows you to 'distribute' the multiplication over each term within the parentheses. For instance, the expression \(a(b+c)\) can be simplified by multiplying \(a\) by \(b\) and \(a\) by \(c\), resulting in \(ab + ac\).

Let's apply this to our example: \(3(2m+4(n+5p))\). Here, we need to distribute the 3 over \(2m\) and \(4(n+5p)\). This gives us \(6m\) from \(3 \times 2m\) and \(12n + 60p\) from \(3 \times 4n\) and \(3 \times 20p\). The distributive property helps break down expressions into more manageable pieces before further simplification.

Once you have distributed the coefficients over each term using the distributive property, the next step in simplifying algebraic expressions is to combine like terms. This process involves adding or subtracting terms that have the same variable raised to the same power. The coefficients of these terms are added or subtracted, while the variable part remains unchanged.

In our example, after distribution, we get \(6m + 12n + 60p + 6n\). Here \(12n\) and \(6n\) are like terms, as they both have the variable \(n\) raised to the first power. We add their coefficients together to get \(18n\) and rewrite the expression as \(6m + 18n + 60p\), a much simpler form. Remember, only like terms can be combined – terms with different variables or exponents must remain separate.

Algebraic simplification is the process of making an expression as straightforward as possible. This often involves applying the distributive property and combining like terms but can also include other methods such as factoring, canceling terms, and reducing fractions when applicable. The goal is to write the expression in the most condensed form without changing its value.

In our exercise, after using the distributive property and combining like terms, we simplify \(3(2m+4(n+5p))+6n\) down to \(6m + 18n + 60p\). This final expression is easy to interpret and use in any further calculations. Remember that simplification can make solving equations, graphing functions, and understanding relationships between variables much more straightforward. It is an essential skill in algebra that enables students to tackle more complex problems.