Problem 22 - Algebra for College Students

Question

Find each product. $$(-2 x)\left(-6 x^{3}\right)\left(x^{2}\right)$$

Step-by-Step Solution

Answer

The product is \(12x^6\).

Step 1: Multiply the coefficients

To find the product, first multiply the numeric coefficients of the terms in the expression \((-2)\), \(-6\), and the implicit \(1\) from \(x^2\). The calculation is: \((-2) \times (-6) \times 1 = 12\).

Step 2: Multiply the variables

For the variables, use the rule \(x^m \times x^n = x^{m+n}\). Identify the exponents of \(x\) in each term: \(x^1\) from \(-2x\), \(x^3\) from \(-6x^3\), and \(x^2\). Summing the exponents gives: \(1 + 3 + 2 = 6\). So, multiplying the variables results in \(x^6\).

Step 3: Write the final product

Combine the results from Step 1 and Step 2 to express the final product as \(12x^6\).

Key Concepts

Multiplication of Variables

In algebra, multiplying variables involves dealing with letters that represent numbers. When we multiply variables, especially when they are similar, there is a straightforward rule we can follow. This is:

• Combine like terms: When you have variables that are the same, such as multiple instances of \(x\), you simply add their exponents. For example, when you multiply \(x^1\) and \(x^2\), you get \(x^{1+2} = x^3\).
• Handle them separately first: In problems like \((-2x)(-6x^3)(x^2)\), group the variables together and coefficients separately. This makes calculations simpler and helps avoid errors.

Since the base (\(x\)) is the same, you only need to focus on the exponents. Adding exponents simplifies multiplication of variables. So, our terms \(-2x\), \(-6x^3\), and \(x^2\) combined would give: \(x^{1+3+2} = x^6\). This rule is a building block for understanding how to work with algebraic expressions effectively.

Exponents in Algebra

Exponents represent how many times a number, called the base, is multiplied by itself. Consider something like \(x^3\). This is equivalent to saying \(x \, \times \; x \, \times \, x\). When multiplying powers with the same base, you'll need to add their exponents together.

• Same base rule: For example, \(x^a \times x^b = x^{a+b}\). This is a fundamental property when working with exponents in algebra.
• Easier calculations: This rule simplifies calculations and keeps equations neat. When you have many terms, adding those exponents is usually simpler than multiplying everything out completely.

Let's apply this to our earlier expression: \(x^1\), \(x^3\), and \(x^2\) become \(x^{1+3+2} = x^6\). Recognizing these rules helps you become adept at handling mathematical expressions quickly and accurately, and is crucial for solving algebraic problems efficiently.

Product of Coefficients

In algebra, coefficients are the numerical part of terms in an algebraic expression. They can be seen as the numbers in front of variables or terms without variables themselves.

• First, focus on numbers: Coefficients are multiplied together separately from the variables. For example, when given an expression like \((-2x)(-6x^3)(x^2)\), you first focus on the numerical coefficients: \(-2\), \(-6\), and the implicit \(1\) from \(x^2\).
• Calculate step-by-step: Multiply them: \((-2) \times (-6) \times 1 = 12\). Performing multiplication in steps helps prevent errors and keeps calculations organized.

The result is purely numerical and serves as a multiplier for any variable results you calculate separately. Mastering the ability to identify and correctly calculate products of coefficients is a crucial skill in algebra, facilitating accurate and efficient problem solving across various algebraic tasks.