Problem 21 - Elementary and Intermediate Algebra

Question

Multiply. See Example 1. $$ \left(2 x^{2} y^{3}\right)\left(4 x^{3} y^{2}\right) $$

Step-by-Step Solution

Answer

\(8x^5y^5\)

Step 1: Identify Components in Each Expression

First, we'll identify the terms in each expression. The first expression is \(2x^2y^3\) and the second is \(4x^3y^2\). Each expression contains a numeric coefficient and variables with exponents.

Step 2: Multiply the Numeric Coefficients

Now, multiply the numeric coefficients of each expression. For the expression \(2x^2y^3\) the coefficient is 2, and for the expression \(4x^3y^2\) it is 4. Thus, the multiplication is \(2 \times 4 = 8\).

Step 3: Apply the Product Rule to Same Base Variables

For variables with the same base, add their exponents. Multiply the \(x\) terms: \(x^2\) and \(x^3\). Using the rule \(x^a \times x^b = x^{a+b}\), we have \(x^{2+3} = x^5\). Similarly, for the \(y\) terms: \(y^3\) and \(y^2\), it becomes \(y^{3+2} = y^5\).

Step 4: Combine All Components

Combine the results from the previous steps. The coefficients, \(8\), multiply together, and the variable results, \(x^5\) and \(y^5\), combine to form the final expression: \(8x^5y^5\).

Key Concepts

Exponents Rules

Exponents rules form the backbone of simplifying expressions that contain powers. These rules help us manage and operate on numbers with exponents efficiently. The key focus here is on understanding how to manipulate terms where the same base appears more than once in a mathematical expression.
One of the primary rules is the **Product Rule** for exponents, which states that when you multiply two powers with the same base, you add their exponents.

• For example, if you have bases of **x** raised to powers **a** and **b**, the product rule is expressed as: \( x^a \times x^b = x^{a+b} \).

This rule makes it possible to combine like terms with the same variable base, ensuring cleaner, more manageable expressions. Applying these rules appropriately simplifies complex expressions, making it easier to solve equations in algebra.

Polynomial Multiplication

Polynomial multiplication involves expanding the product of two polynomials or expressions. Each term in the first polynomial must be distributed across every term in the second polynomial. This uses the distributive property to ensure each term is multiplied correctly and completely.
Consider the multiplication of \( (2x^2y^3)(4x^3y^2) \).
Here, each term from the first polynomial **(2x²y³)** is multiplied by each term from the second polynomial **(4x³y²)**.

• Start by multiplying the coefficients: 2 and 4 to get 8.
• Then, apply the product rule for **x** and **y** variables, yielding \( x^{2+3} = x^5 \) and \( y^{3+2} = y^5 \).

While multiplying polynomials, maintain careful organization of terms to avoid mistakes, ensuring an accurate final expression that is both expanded and simplified.

Product Rule for Exponents

The Product Rule for Exponents is a critical rule when multiplying terms that have identical bases.
This rule simplifies the multiplication by letting you add the exponents for the same base instead of manually expanding the multiplication.For example, when using the Product Rule:

• If we have two powers, \( x^2 \) and \( x^3 \), the product \( x^2 \times x^3 \) becomes \( x^{2+3} = x^5 \).
• Similarly, for \( y^3 \times y^2 \), you get \( y^{3+2} = y^5 \).

The rule emphasizes that as long as the bases are the same, the exponents can be seamlessly added together, regardless of the specific numbers involved. This not only simplifies calculations but also helps in keeping expressions clear and concise in algebra.