Simplify The Expression 2a³(a + 3b²) - 6b²(2a³ - 1) A Step-by-Step Guide

by ADMIN 73 views

In the realm of mathematics, simplifying expressions is a fundamental skill. This article will delve into simplifying the expression 2a³(a + 3b²) - 6b²(2a³ - 1). We will break down each step, providing a clear and concise explanation to help you understand the process thoroughly. Mastering this technique is crucial for solving more complex algebraic problems and lays the groundwork for advanced mathematical concepts. Our goal is to not only simplify the given expression but also to equip you with the knowledge and confidence to tackle similar problems independently. So, let's embark on this mathematical journey together and unravel the intricacies of simplifying expressions.

Understanding the Basics of Algebraic Expressions

Before we dive into the specifics of our expression, let's establish a firm understanding of the basic principles governing algebraic expressions. An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables are symbols (usually letters) that represent unknown quantities, while constants are fixed numerical values. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations must be performed to correctly simplify an expression.

The Distributive Property

The distributive property is a cornerstone of simplifying expressions, especially those involving parentheses. It states that for any numbers a, b, and c, a(b + c) = ab + ac. This means that the term outside the parentheses must be multiplied by each term inside the parentheses. Understanding and applying the distributive property correctly is paramount for accurate simplification. This property allows us to remove parentheses and combine like terms, paving the way for a more simplified form of the expression. In our example, we will heavily rely on the distributive property to expand the terms and eliminate the parentheses.

Combining Like Terms

Combining like terms is another essential technique in simplifying expressions. Like terms are terms that have the same variables raised to the same powers. For instance, 3x² and 5x² are like terms, while 3x² and 5x are not. We can combine like terms by adding or subtracting their coefficients (the numerical part of the term). This process helps to reduce the number of terms in the expression, making it more concise and easier to work with. Identifying and combining like terms effectively is crucial for achieving the simplest form of the expression. This step often follows the application of the distributive property and plays a vital role in the overall simplification process.

Step-by-Step Simplification of the Expression

Now, let's apply these foundational principles to simplify the given expression: 2a³(a + 3b²) - 6b²(2a³ - 1). We will proceed step-by-step, explaining each action and the reasoning behind it. This methodical approach will not only lead us to the correct answer but also reinforce your understanding of the simplification process.

1. Applying the Distributive Property

Our first step is to apply the distributive property to both terms in the expression. We need to multiply 2a³ by each term inside the first parentheses (a + 3b²) and -6b² by each term inside the second parentheses (2a³ - 1). This process will eliminate the parentheses and create individual terms that can be further simplified.

  • 2a³(a + 3b²) = 2a³ * a + 2a³ * 3b² = 2a⁴ + 6a³b²
  • -6b²(2a³ - 1) = -6b² * 2a³ + (-6b²) * (-1) = -12a³b² + 6b²

By applying the distributive property, we have successfully expanded the expression and removed the parentheses. We now have a series of individual terms that can be further simplified by combining like terms.

2. Combining Like Terms

After applying the distributive property, our expression looks like this: 2a⁴ + 6a³b² - 12a³b² + 6b². Now, we need to identify and combine like terms. Remember, like terms have the same variables raised to the same powers. In this expression, 6a³b² and -12a³b² are like terms because they both contain the variables a³ and b².

Combining these like terms, we get:

  • 6a³b² - 12a³b² = -6a³b²

Now, we substitute this result back into the expression, which gives us:

  • 2a⁴ - 6a³b² + 6b²

We have successfully combined the like terms, further simplifying the expression. This step is crucial for arriving at the most concise and manageable form of the expression.

3. The Simplified Expression

After applying the distributive property and combining like terms, we have arrived at the simplified expression: 2a⁴ - 6a³b² + 6b². There are no more like terms to combine, and the expression is in its simplest form. This is our final answer.

Common Mistakes to Avoid

Simplifying expressions can be tricky, and it's easy to make mistakes if you're not careful. Let's look at some common pitfalls to avoid:

Incorrectly Applying the Distributive Property

A frequent error is failing to distribute the term outside the parentheses to every term inside. Remember, each term inside the parentheses must be multiplied by the term outside. Double-check your work to ensure you've distributed correctly.

Mixing Up Signs

Sign errors are common, especially when dealing with negative numbers. Pay close attention to the signs when applying the distributive property and combining like terms. A simple sign mistake can throw off the entire calculation.

Combining Unlike Terms

Only like terms can be combined. Avoid the mistake of adding or subtracting terms that have different variables or different powers. Ensure that the variables and their exponents are identical before combining terms.

Forgetting the Order of Operations

Always follow the order of operations (PEMDAS/BODMAS). Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Ignoring this order can lead to incorrect simplifications.

Practice Problems

To solidify your understanding of simplifying expressions, here are some practice problems for you to try:

  1. 3x(2x - 5) + 4(x² + 2)
  2. 5y²(y - 3) - 2y(y² + 4y)
  3. (4a + 2b)(a - 3b)
  4. 7m³(2m² - m + 3) - 5m(m⁴ + 2m²)

Working through these problems will reinforce your skills and help you identify areas where you may need further practice.

Conclusion

Simplifying the expression 2a³(a + 3b²) - 6b²(2a³ - 1) involves applying the distributive property and combining like terms. By following these steps carefully, we can arrive at the simplified form: 2a⁴ - 6a³b² + 6b². Mastering this process is crucial for success in algebra and beyond. Remember to avoid common mistakes, practice regularly, and approach each problem systematically. With consistent effort, you'll become proficient in simplifying expressions and tackling more complex mathematical challenges.