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The distributive property is a fundamental concept in algebra that helps break down difficult expressions into manageable parts. The core idea is to multiply a term on the outside of a parenthesis with each term inside the parenthesis.

For example, in the expression \(5(xy + 2)\), the number 5 outside the parentheses multiplies both \(xy\) and 2. This step-by-step breakdown results in:

• \(5 \times xy = 5xy\)
• \(5 \times 2 = 10\)

After applying the distributive property, the expression becomes \(5xy + 10\). This method is especially helpful in simplifying expressions, which is a vital step in solving equations. It allows for redistributing terms throughout various parts of an equation, simplifying calculations, and making complex expressions more understandable. It’s a powerful tool to have in your algebraic toolkit, helping to transform daunting expressions into clear, solvable elements.

Once the distributive property has been applied, the next important step in simplifying algebraic expressions is combining like terms. Like terms refer to the terms in an expression that have identical variable parts. This means that the variable is the same, and its exponent is the same too.

In our example:

• From \(5xy + 10 - 3xy - 8\), the like terms are \(5xy\) and \(-3xy\). They both contain the variable part \(xy\).
• The numbers, 10 and -8, are also like terms because they are constants and don’t involve variables.

By combining them, we simplify the expression into \(2xy + 2\):

• Combine \(5xy\) and \(-3xy\) to get \(2xy\).
• Combine 10 and -8 to get 2.

Combining like terms helps streamline the expression, reducing it into a simpler form that is easier to handle and understand. This method is fundamental in algebra because it bridges the gap between complex and simpler expressions.

Algebraic simplification is about making an expression as simple as possible. It involves using algebraic techniques such as the distributive property and combining like terms to reduce the number of terms and make calculations more straightforward.

Let's simplify the expression step by step:

• First, apply the distributive property: \(5(xy + 2)\) becomes \(5xy + 10\).
• Subtract \(3xy\) and 8: it becomes \(5xy + 10 - 3xy - 8\).
• Then, combine like terms: the \(xy\) terms \(5xy\) and \(-3xy\) simplify to \(2xy\). The constants 10 and -8 simplify to 2.

This ends with a simplified expression of \(2xy + 2\).

Simplifying expressions not only makes them easier to understand but is crucial when solving equations as it can significantly reduce the complexity of problems. By learning to simplify effectively, students can tackle more challenging problems with confidence and precision.