Problem 35 - Elementary Algebra Within Reach

Question

In Exercises 19-36, expand the expression as a product of factors. $$ \left[2(a-b)^{3}\right]\left[2(a-b)^{2}\right] $$

Step-by-Step Solution

Answer

The expanded expression as a product of factors is \(4(a-b)^{5}\).

Step 1: Identify components

Firstly, identify the components of the expression. We have \(2(a-b)^{3}\) and \(2(a-b)^{2}\).

Step 2: Apply the multiplication rule of exponents

Use the multiplication rule of exponents which states that when multiplying like bases, you simply add the exponents. This means that \( (a-b)^{3} \times (a-b)^{2} = (a-b)^{5} \)

Step 3: Combine multipliers

The expression also has 2 from each part, we combine those as well. This gives us 2*2 = 4.

Step 4: Apply multiplication

Now multiply the combined 4 with the simplified base and exponent which gives us the final expression of \(4(a-b)^{5}\).

Key Concepts

Multiplication Rule of Exponents

Understanding the multiplication rule of exponents is essential when expanding expressions involving powers. This rule states that when you multiply two expressions with the same base, you can add the exponents. For instance, if you have a term like \( (a-b)^{3} \) multiplied by another term with the same base, such as \( (a-b)^{2} \), rather than multiplying each term inside the parentheses, you simply add the exponents to get \( (a-b)^{3+2} \), which simplifies to \( (a-b)^{5} \).

This is a fundamental concept in algebra that helps streamline the process of working with exponential expressions. Remember, this rule only applies when the bases are identical. Misapplying this rule can lead to incorrect solutions,
so emphasizing its correct use is pivotal. By applying this rule to the given exercise, we were able to quickly identify that \(2(a-b)^{3} \) and \( 2(a-b)^{2} \) can be combined to \( 4(a-b)^{5} \) simplifying the original problem significantly.

Factoring Algebraic Expressions

Factoring algebraic expressions is like breaking down a complex number into its building blocks. When looking at a term like \(2(a-b)^{3} \) you should see it as a combination of a numerical factor (2) and an algebraic factor \( (a-b)^{3} \).

Factoring becomes essential when dealing with the simplification of algebraic expressions or solving equations. It involves looking for common factors in terms to simplify or solve them effectively. In our example, we noted that there's a common numerical factor of 2 in both expressions. Recognizing common factors allows you to combine and simplify expressions,
which is a step toward factoring more complex algebraic expressions. When we combine our numerical factors (2 from each term), we effectively factored out a 4 from the original expression.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a way to make complex expressions easier to understand and work with. In the context of the provided exercise, simplification is done by applying the multiplication rule of exponents and combining like terms. After combining the exponents, we also combined the numeric multipliers (2) from each expression, resulting in the number 4.

The aim is to get the simplest form possible to make it visually less complex and easier to manipulate in further operations or equations. Simplification can involve other steps, such as reducing fractions or collecting like terms, but in the context of our exercise, it was a matter of applying basic exponent rules and arithmetic. The final simplification of the given expressions yielded a much cleaner result: \(4(a-b)^{5} \) which is the product of a numerical and an algebraic factor. This process aids in solving more complex algebraic problems and makes it easier for students to grasp the underlying concepts.