Problem 30 - Algebra for College Students

Question

Find each product. $$(3 b)\left(-2 a b^{2}\right)(7 a)$$

Step-by-Step Solution

Answer

The product is \(-42a^2b^3\).

Step 1: Simplify Each Term

To find the product, first look at each term individually: - The first term is \(3b\).- The second term is \(-2ab^2\).- The third term is \(7a\). This step involves identifying the multiplicands, which include numbers, the variable \(a\), and the variable \(b\).

Step 2: Multiply the Numerical Coefficients

Multiply the numerical coefficients from each term: - The coefficients are 3, -2, and 7.- Multiply them: \[3 \times (-2) \times 7 = -42\]

Step 3: Multiply the Variable 'a'

The variable \(a\) appears in two of the terms: - \(-2ab^2\) has one \(a\), and \(7a\) has one \(a\).- Multiply them: \[a^1 \times a^1 = a^{1+1} = a^2\]

Step 4: Multiply the Variable 'b'

The variable \(b\) appears in two of the terms:- \(3b\) has one \(b\) and \(-2ab^2\) has two \(b\)'s. - Multiply them together: \[b^1 \times b^2 = b^{1+2} = b^3\]

Step 5: Combine Everything Together

Combine the results from the previous steps: - The numerical coefficient is \(-42\).- The \(a\) term is \(a^2\).- The \(b\) term is \(b^3\).- The product is: \[-42a^2b^3\]

Key Concepts

Multiplying Terms

In algebra, multiplying terms involves combining numbers (coefficients) and variables. When multiplying terms, each part of the terms—the coefficient and the variable(s)—needs to be considered separately and then combined together. This is important because violating the order can lead to incorrect results.
One crucial aspect of multiplying terms is understanding when each variable appears in the terms involved. Variables that appear more than once get their exponents added together rather than multiplied. This is because the operation of multiplying adds the powers as per the laws of exponents.
Let's look at a practical example: multiplying the terms \(3b(-2ab^2)(7a)\). For these three separate terms:

• The numerical coefficients are 3, -2, and 7.
• The variables involved are 'a' and 'b'.

The key is to handle each component separately before putting them back together. This practice helps in visualizing and clearly understanding the multiplication process.

Numerical Coefficients

Numerical coefficients are the numerical parts of algebraic terms. When you're working with algebraic expressions, these coefficients are the numbers placed in front of the variables. These coefficients are essential because they dictate the magnitude of the term.
For instance, in the product we are considering \(3b(-2ab^2)(7a)\), the coefficients are 3, -2, and 7. To multiply these coefficients:

• First, multiply the first two: \(3 \times -2 = -6\).
• Then, multiply the result by the next coefficient: \(-6 \times 7 = -42\).

Organizing your multiplication of numerical coefficients first can simplify the entire process before you involve the variables, and having the numerical result first lays the groundwork for combining it later with the products of variables.

Variables in Algebra

Variables in algebra represent unknown or any number and are typically denoted by letters like 'x', 'y', 'z'. These letters hold the position of numbers and can be exponentiated. When multiplying terms that carry variables, it follows certain rules involving exponents.
When variables with the same base are multiplied together, you simply add their exponents together. This is based on the exponent rule: \(x^m \times x^n = x^{m+n}\). Using this rule, let’s multiply the variables in our example:

• Both the terms \(-2ab^2\) and \(7a\) contain the variable \(a\): \(a^1\) and \(a^1\), combine them to get \(a^{1+1} = a^2\).
• For the variable \(b\), the terms \(3b\) and \(-2ab^2\) have \(b^1\) and \(b^2\), respectively, resulting in \(b^{1+2} = b^3\).

Working with each type of variable individually and using exponent rules streamlines the process of manipulating algebraic expressions. Understanding these rules assists in systematically approaching any algebraic multiplication problem.