Problem 12 - Algebra for College Students

Question

Find each product. $$\left(-8 a^{2} b^{2}\right)\left(-12 a b^{5}\right)$$

Step-by-Step Solution

Answer

96a^3b^7

Step 1: Understand the Expression

The expression is a product of two terms: \(-8a^2b^2\) and \(-12ab^5\). Each term is a monomial, meaning it contains constants and variables raised to powers.

Step 2: Multiply the Coefficients

First, we will multiply the numerical coefficients. The coefficients are \(-8\) and \(-12\). Multiply them together to get \((-8) \times (-12) = 96\).

Step 3: Multiply the Variables with the Same Base

For any two variables with the same base, add their exponents according to the property of exponents \(x^m \times x^n = x^{m+n}\). The variables are \(a\) and \(b\):- For \(a\), add the exponents: \(2 + 1 = 3\).- For \(b\), add the exponents: \(2 + 5 = 7\).

Step 4: Write the Final Product

Combine the results from the previous steps. Multiply the coefficients, and write the variables with the new exponents:\[96a^3b^7\]

Key Concepts

Exponents

In mathematics, exponents are a powerful tool that allows us to express how many times a number, known as the base, is used as a factor in repeated multiplication. In the context of algebra, exponents are critical to understanding how to handle variables. For example, in the expression \(a^2\), the "2" is the exponent, indicating that \(a\) is multiplied by itself: \(a \times a\).
When multiplying terms with the same base, the exponents can be simplified by adding them together. This rule is formalized as:

• \( x^m \times x^n = x^{m+n} \)

For instance, if you have \(a^2\) and \(a^1\), you can combine them as \(a^{2+1} = a^3\). Understanding how to manipulate exponents in this way simplifies complex algebraic expressions and problems.

Monomials

A monomial is a term in algebra that is a product of a constant and any number of variables raised to non-negative integer powers. A simple example is \(-8a^2b^2\). Here, everything together forms a single term – the monomial.
Key characteristics of monomials include:

• It contains no addition or subtraction within its structure.
• It consists of variables and coefficients.
• The variables can have exponents, and these exponents are always whole, non-negative numbers.

Understanding monomials is crucial because they are the basic building blocks of expressions and equations in algebra.

Coefficients

Coefficients are the numerical parts of the terms in an algebraic expression. They multiply the variables, giving the term its overall magnitude. In the expression \(-8a^2b^2\), \(-8\) is the coefficient, while in \(-12ab^5\), \(-12\) serves as the coefficient.
To solve problems involving polynomials, coefficients are handled separately from the variables. In the multiplication of monomials like in our exercise, the coefficients \(-8\) and \(-12\) are multiplied separately to yield \(96\). Paying attention to coefficients ensures that algebraic simplifications are accurate.

Variables

Variables are symbols in mathematics, often letters like \(a\) and \(b\), that represent unknown or general quantities. They bring flexibility to equations and expressions, allowing them to model real-world situations and mathematical relationships.
In our original exercise, the variables \(a\) and \(b\) are each raised to certain powers, and this influences their behavior when multiplied. Maintaining correct order and applying the laws of exponents to variables, such as \(a^2\times a^1 = a^3\) and \(b^2\times b^5 = b^7\), is essential for the accurate combination of terms. Comprehending how variables work within expressions is fundamental to solving polynomial multiplication and other algebraic operations.