Problem 27 - Algebra 2

Question

Simplify. $$ -5 a b^{2}\left(-3 a^{2} b+6 a^{3} b-3 a^{4} b^{4}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(15 a^3 b^3 - 30 a^4 b^3 + 15 a^5 b^6\).

Step 1: Distribute the Term

We need to distribute \(-5ab^2\) into each term within the parentheses. The expression is: \- \[-5ab^2(-3a^2b + 6a^3b - 3a^4b^4)\]. We will multiply term by term. Start with the first term: \(-5ab^2 \times -3a^2b\).

Step 2: Distribute to the First Term

Multiply \\(-5ab^2\) by \\(-3a^2b\):\ \(-5 \times -3 = 15\), \(a^{1+2} = a^3\), and \\(b^{2+1} = b^3\).So, the product is \\(15 a^3 b^3\).

Step 3: Distribute to the Second Term

Now multiply \\(-5ab^2\) by \\(6a^3b\):\(-5 \times 6 = -30\), \\(a^{1+3} = a^4\), and \\(b^{2+1} = b^3\).Thus, the product is \\(-30 a^4 b^3\).

Step 4: Distribute to the Third Term

Multiply \\(-5ab^2\) by \\(-3a^4b^4\):\(-5 \times -3 = 15\), \\(a^{1+4} = a^5\), and \\(b^{2+4} = b^6\).This gives \\(15 a^5 b^6\).

Step 5: Combine All Terms

Combine the products from Steps 2, 3, and 4:\(15 a^3 b^3 - 30 a^4 b^3 + 15 a^5 b^6\).This expression is fully simplified as there are no like terms to combine further.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It helps simplify expressions by distributing a term across terms inside parentheses. In simple words, it allows you to multiply a single term by each term inside a set of parentheses. For instance, when you have an expression like \(-5ab^2(-3a^2b + 6a^3b - 3a^4b^4)\), you can distribute the term \(-5ab^2\) to each of the terms within the parentheses.

The main point to remember is that multiplication distributes over addition and subtraction, which means:

• Multiply the outside term with each term inside the parentheses separately.
• Apply the sign (positive or negative) of the outside term to each product.

This technique is particularly useful when dealing with algebraic expressions, making them simpler to work with by breaking them down into manageable parts.

Exponent Rules

Exponent rules, also known as laws of exponents, are essential when working with polynomial expressions. They allow you to simplify terms that involve powers of variables. Here are a few key rules that were applied:

• Product of Powers Rule: When multiplying terms with the same base, you add the exponents. For example, \(a^m \cdot a^n = a^{m+n}\).
• Power of a Product Rule: When a power is applied to a product, it applies to each factor within. So, \((ab)^n = a^n \cdot b^n\).
• Basic Multiplication: Even outside of exponents, ensure numerical coefficients are multiplied. Hence, \(-5 \times -3 = 15\).

By applying these rules, you can easily handle the products of terms like \(-5ab^2\) times \(-3a^2b\) by aligning similar bases and combining their exponents.

Algebraic Expressions

Algebraic expressions are a core component of algebra. They are used to represent mathematical phrases and contain numbers, variables, and operations. Understanding how to manipulate these expressions is crucial for solving equations and simplifying terms.

For instance, consider the expression \(15 a^3 b^3 - 30 a^4 b^3 + 15 a^5 b^6\). This is a standard form where algebraic terms are combined through addition or subtraction. Noticeably, each term involves coefficients (numbers), variables \(a\) and \(b\), and their exponents.

One of the goals when simplifying algebraic expressions is to combine like terms. "Like terms" are terms that have the same variables raised to the same powers, which can be added or subtracted together. In our example, none of the terms can be combined further as their variable powers differ.

By understanding and recognizing the components of algebraic expressions, you gain the ability to convert complex expressions into manageable outcomes, ready for further operations or interpretations.