Problem 6 - Prealgebra

Question

Find each product or quotient. Express using exponents. $$-3 x^{2}\left(4 x^{3}\right)$$

Step-by-Step Solution

Answer

The product is \(-12x^{5}\).

Step 1: Identify the Components

Look at the expression \(-3 x^{2}(4 x^{3})\). The components consist of coefficients \(-3\) and \(4\), and the variables with exponents \(x^{2}\) and \(x^{3}\). Our goal is to find the product of these parts.

Step 2: Multiply the Coefficients

First, multiply the coefficents \(-3\) and \(4\): \(-3 \times 4 = -12\).

Step 3: Multiply the Variables Using Exponent Rules

Apply the rule of exponents which states that when you multiply like bases, you add the exponents \(a^{m} \times a^{n} = a^{m+n}\). Here, multiply \(x^{2}\) and \(x^{3}\): \(x^{2+3} = x^{5}\).

Step 4: Form the Final Expression

Combine the results from Step 2 and Step 3. The product of the expression is now \(-12x^{5}\).

Key Concepts

Multiplication of Coefficients

When dealing with algebraic expressions like \(-3 x^{2}(4 x^{3})\), it's crucial to first understand how to handle the coefficients. Coefficients are the numerical parts of terms, separate from the variables. Here, \(-3\) and \(4\) are the coefficients. Multiplying coefficients involves basic arithmetic: you simply multiply the numerical values together.

In this case, you multiply \(-3\) by \(4\) to get \(-12\). The negative sign is important too; remember a negative multiplied by a positive is always negative. Therefore, this step simplifies the part of the expression with numbers.

• Identify the numerical coefficients.
• Multiply them directly: \(-3 \times 4 = -12\).

This step deals only with numbers, setting the stage for handling the variable part of the problem.

Exponents Addition

When multiplying expressions with the same base, exponents play a critical role. The rule for multiplying exponents is simple. If you have \(x^{m} \times x^{n}\), then you add the exponents: \(x^{m+n}\).

In our example, \(x^{2}\) and \(x^{3}\) are the variable parts. Since they share the same base, \(x\), you add the exponents together: \(x^{2+3} = x^{5}\). This application of the exponent rule makes managing the algebraic expression much simpler.

• Ensure the bases are the same before applying the rule.
• Add the exponents: \(2 + 3 = 5\).

By systematically adding exponents, you maintain the integrity of the mathematical operation.

Algebraic Expressions

Combining the multiplication of coefficients and the addition of exponents results in an expression known as an algebraic expression. An algebraic expression can contain numbers, variables, and exponents.

Our task was to simplify: \(-3 x^{2}(4 x^{3})\). After applying the multiplication of coefficients and the addition of exponents, the final expression becomes \(-12x^{5}\).

• Start by simplifying individual components.
• Combine the results to form a cohesive expression.

This process of simplification takes expressions that may initially seem complex and translates them into manageable, straightforward forms. By understanding each component's role, the wider algebraic operation comes into focus, reinforcing your grasp on foundational algebraic principles.