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The distributive property is a fundamental concept in algebra that simplifies expressions and calculations by distributing one quantity across others inside parentheses. When you encounter an expression like \((-3ab^2)(5ab)\), you can think of it as distributing or multiplying each term inside one set of parentheses with the term from the other set.

To apply this property, you multiply the coefficients first, and then the variables separately:

• Multiply \(-3\) by \(+5\) to get \(-15\).
• Combine the multiplication of each variable, respecting their exponents.

This approach allows you to manage complicated expressions and simplifies calculations, especially when handling polynomials. It makes sure that each component of one term is applied to each component of the other term, ensuring no part is overlooked. Proper application leads to easier and more reliable simplification of expressions.

Exponents play a crucial role in simplifying expressions with variables. The rules of exponents help you manage terms with powers when multiplying expressions.

In our exercise, consider the term \(b^2 imes b\). When you multiply similar bases (the 'b' here), you add their exponents. This gives you:

• \(b^2 imes b = b^{2+1} = b^3\)

Understanding this rule helps when dealing with polynomials or any algebraic expressions since it simplifies combining powers. Knowing that you just add the exponents when the bases are the same is a powerful tool to break down problems, making them more manageable and less intimidating. This is essential knowledge for efficient algebraic manipulation and solving.

Coefficient multiplication involves multiplying the numerical parts (coefficients) of terms in an algebraic expression. It's straightforward but crucial for simplifying expressions or when using the distributive property.

In our problem, \(-3 imes 5\) was the coefficient multiplication step resulting in \(-15\). Here’s why it's important:

• It simplifies the expression to make it easily understandable.
• Ensures consistency when further combining terms (like variables).

When you multiply coefficients, you maintain the operation sign (positive or negative), which affects the result's overall sign. Overlooking the role of the sign can lead to incorrect simplification, so always pay attention to both numeric value and sign during multiplication. Efficient handling of coefficients is a foundational skill that supports more complex operations in algebra.