Problem 51 - Intermediate Algebra

Question

Perform the indicated operations. \(\left(2 x^{3}\right)\left(-4 x^{2}\right)\)

Step-by-Step Solution

Answer

The product is \(-8x^5\).

Step 1: Understanding the Problem

We need to perform the multiplication between two monomials: \(2x^3\) and \(-4x^2\). This requires multiplying both the coefficients and the powers of the variable \(x\).

Step 2: Multiply the Coefficients

First, multiply the numerical coefficients of the two monomials: \(2\) and \(-4\). \[ 2 \times (-4) = -8 \]

Step 3: Apply the Power of a Product Rule

Multiply the powers of \(x\) by adding the exponents of the two \(x\) terms: \(x^3\) and \(x^2\). According to the power of a product rule, we add the exponents: \[ x^{3+2} = x^5 \]

Step 4: Combine the Results

Combine the results of the multiplied coefficients and the summed exponents of \(x\): Thus, our final result is \[ -8x^5 \]

Key Concepts

coefficients multiplication

When multiplying monomials, the first step is to focus on the numeric values, or coefficients, within each monomial. In the given problem, we have coefficients of 2 and -4 from the monomials \(2x^3\) and \(-4x^2\), respectively. Let's break down how to multiply these coefficients:

• Identify the coefficients: \(2\) for the first monomial and \(-4\) for the second.
• Multiply the coefficients together: \(2 imes (-4)\).
• Calculate the product, which is \(-8\).

The key point here is to handle the signs correctly while multiplying the numbers. When you multiply a positive number by a negative number, the result is negative, hence the outcome is \(-8\). Now that the coefficients are multiplied, we can shift focus to the variables.

power of a product rule

Next, you'll want to apply the power of a product rule to handle the variables. Each monomial in our exercise features the variable \(x\), with different exponents. The power of a product rule helps to simplify variable multiplication:

• Identify the exponents: For \(2x^3\), the exponent is 3, and for \(-4x^2\), the exponent is 2.
• Apply the power of a product rule, which states that when you multiply like bases, you add their exponents.
• Add the exponents together: \(3+2=5\).
• Apply this result to the variable, yielding \(x^5\).

This rule is a standard for simplifying expressions with variables having the same base. Hence, after applying the rule, we have simplified the expression involving \(x\) to \(x^5\), ready for combination with the coefficient multiplication result.

adding exponents

Adding exponents is an essential process when you're multiplying terms that share the same base. This concept allows you to combine powers efficiently using the power of a product rule as discussed earlier. Here’s a closer look at the steps involved in our exercise:

• Each monomial has a base of \(x\), specifically \(x^3\) from \(2x^3\) and \(x^2\) from \(-4x^2\).
• When multiplying these, the bases remain the same and you perform addition on the exponents: \[ x^3 imes x^2 = x^{3+2} \].
• This results in \(x^5\), which shows the base \(x\) raised to the power 5 after operations.

Adding the exponents directly is a simplified way to handle product problems involving powers and ensures accuracy in combining like bases. The idea is that as long as the multiplied bases are the same, exponents can always be directly summed.