How To Simplify The Algebraic Expression -a^3(ab+b^2)

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It allows us to manipulate complex equations into more manageable forms, making them easier to understand and solve. Among the various algebraic operations, the distributive property plays a crucial role in simplifying expressions involving parentheses. In this comprehensive guide, we will delve into the process of simplifying the given expression: $-a^3(ab+b^2)$. We will break down each step, providing clear explanations and insights along the way, ensuring a thorough understanding of the underlying principles.

Understanding the Distributive Property

The distributive property is a cornerstone of algebra. It states that for any numbers a, b, and c, the following holds true: a(b + c) = ab + ac. In essence, the distributive property allows us to multiply a term outside the parentheses with each term inside the parentheses. This property is not limited to addition; it also applies to subtraction: a(b - c) = ab - ac. Mastering the distributive property is essential for simplifying algebraic expressions effectively.

Applying the Distributive Property to Our Expression

Now, let's apply the distributive property to the expression $-a^3(ab+b^2)$. Here, $-a^3$ is the term outside the parentheses, and $(ab+b^2)$ is the expression inside the parentheses. To simplify, we need to multiply $-a^3$ with each term inside the parentheses:

$-a^3(ab+b^2) = (-a^3)(ab) + (-a^3)(b^2)$

This step involves distributing $-a^3$ to both $ab$ and $b^2$. Next, we perform the multiplications:

$(-a^3)(ab) = -a^3 * a * b = -a^(3+1) * b = -a^4 b$

$(-a^3)(b^2) = -a^3 * b^2$

Combining the Terms

Now that we have multiplied $-a^3$ with each term inside the parentheses, we can combine the resulting terms:

$-a^4 b + (-a^3 b^2) = -a^4 b - a^3 b^2$

Therefore, the simplified expression is $-a^4 b - a^3 b^2$.

Step-by-Step Breakdown of the Simplification Process

To further clarify the process, let's break down the simplification step-by-step:

1. Identify the expression: $-a^3(ab+b^2)$
2. Apply the distributive property: $-a^3(ab+b^2) = (-a^3)(ab) + (-a^3)(b^2)$
3. Multiply the terms:
   • $(-a^3)(ab) = -a^4 b$
   • $(-a^3)(b^2) = -a^3 b^2$
4. Combine the terms: $-a^4 b - a^3 b^2$

This step-by-step approach ensures that we follow the correct order of operations and arrive at the accurate simplified expression.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's crucial to be aware of common mistakes that can lead to incorrect results. One common mistake is forgetting to distribute the term outside the parentheses to all terms inside the parentheses. For instance, in our example, failing to multiply $-a^3$ with both $ab$ and $b^2$ would result in an incorrect simplification. Another common mistake is errors in exponent arithmetic. Remember that when multiplying terms with the same base, we add the exponents. For example, $a^3 * a = a^(3+1) = a^4$. A thorough understanding of the distributive property and exponent rules is essential to avoid these mistakes.

Conclusion: Mastering Algebraic Simplification

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics. The distributive property is a powerful tool that allows us to remove parentheses and combine like terms, leading to simpler and more manageable expressions. By understanding and applying the distributive property correctly, we can confidently simplify a wide range of algebraic expressions. Remember to practice regularly and pay close attention to details to avoid common mistakes. With consistent effort, you can master the art of algebraic simplification and excel in your mathematical endeavors.

Based on the simplification process detailed above, the correct answer is:

C. $-a^4 b - a^3 b^2$

Let's analyze why the other options are incorrect:

• A. $-a^4 b^3$: This option incorrectly multiplies the exponents of $b$ within the parentheses, rather than treating $b$ and $b^2$ as separate terms.
• B. $-a^4 b+a^3 b^2$: This option makes a sign error when distributing the $-a^3$ term. The second term should be negative, not positive.
• D. $-a^4 b+b^2$: This option incorrectly simplifies the expression and does not properly apply the distributive property to the second term within the parentheses.
• E. None of these: This option is incorrect because option C is the correct simplification.

While the distributive property is crucial, there are other techniques that can further simplify algebraic expressions. One such technique is combining like terms. Like terms are terms that have the same variables raised to the same powers. For example, in the expression $3x^2 + 2x - x^2 + 5x$, $3x^2$ and $-x^2$ are like terms, and $2x$ and $5x$ are like terms. We can combine like terms by adding or subtracting their coefficients. In this case, $3x^2 - x^2 = 2x^2$ and $2x + 5x = 7x$, so the simplified expression would be $2x^2 + 7x$. Factoring is another powerful technique. Factoring involves expressing an expression as a product of its factors. For example, the expression $x^2 + 5x + 6$ can be factored as $(x + 2)(x + 3)$. Factoring can be useful for simplifying expressions, solving equations, and identifying common factors.

The Importance of Order of Operations

When simplifying algebraic expressions, it's crucial to follow the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order ensures that we perform operations in the correct sequence, leading to accurate results. In our example, we first addressed the parentheses by applying the distributive property. Then, we performed the multiplications and finally combined the terms. Adhering to the order of operations is essential for avoiding errors and simplifying expressions effectively.

Practice Problems for Mastery

To solidify your understanding of simplifying algebraic expressions, practice is key. Here are a few practice problems:

1. Simplify: $2x(3x - 4) + 5(x + 2)$
2. Simplify: $-3y^2(2y + 1) - 4y(y^2 - 3)$
3. Simplify: $(a + b)(a - b) + a^2 + b^2$

Working through these problems will help you apply the concepts we've discussed and build confidence in your algebraic simplification skills.

By mastering these techniques and consistently practicing, you'll be well-equipped to tackle complex algebraic expressions and excel in your mathematical journey. Remember, the key to success lies in understanding the fundamental principles and applying them diligently.