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When we talk about the distributive property in algebra, we are referring to the rule that lets us multiply a single number by a group of numbers inside a parenthesis. It's like sharing candy evenly among friends; everything in the group gets a piece based on the given number. For example, in the expression \(8(2x + 5)\), we use the distributive property to multiply 8 by each term inside the parentheses.

Here's the step-by-step:

• Multiply 8 by \(2x\) to get \(16x\).
• Then, multiply 8 by 5 to get 40.

This gives us \(16x + 40\). The same applies for the second part, where -7 is multiplied by each term inside its parenthesis. Distributing -7 gives us \(-7x + 63\). Distributive property helps break down complex expressions into smaller, more manageable parts, making it easier to handle.

Once we've applied the distributive property, the next step involves simplifying the expression by combining like terms. Like terms are terms that have the same variables raised to the same powers. Specifically, you can combine them because they belong to the same category, like sorting socks of the same color together.

In our example, after distribution, we ended up with two \(x\) terms: \(16x\) and \(-7x\). We also have constant terms: 40 and 63.

• By adding \(16x - 7x\), we get \(9x\).
• By adding the constants, 40 and 63, we get 103.

Combining like terms simplifies the expression by reducing it to the simplest form where the same categories are summed together. This is essential to make problems more manageable and to find solutions easily.

At the heart of algebra is the idea of simplification. Simplifying an expression means reducing it to its most compact, understandable form. This process often involves using the distributive property and combining like terms, as we've seen.

In our exercise, after distributing and combining like terms, the expression \(8(2x+5)-7(x-9)\) was simplified to \(9x + 103\). Simplification isn't just about finding a neat answer; it's about making an expression more useful and easier to understand.

Here are a few reasons why simplification is important:

• It helps in solving equations more efficiently.
• It provides a clearer picture of what the expression actually represents.
• It reduces the complexity of further operations, such as calculus manipulations or solving related problems.

Ultimately, simplification is a critical skill that helps in organizing and clarifying mathematical thoughts.