Problem 19 - Algebra for College Students

Question

Find each product. $$\left(-\frac{1}{2} x y\right)\left(\frac{1}{3} x^{2} y^{3}\right)$$

Step-by-Step Solution

Answer

The product is \(-\frac{1}{6}x^3y^4\).

Step 1: Multiply the Coefficients

First, multiply the coefficients of the two expressions. The coefficients are \(-\frac{1}{2}\) and \(\frac{1}{3}\). Multiply them together: \(-\frac{1}{2} \times \frac{1}{3} = -\frac{1}{6}\).

Step 2: Multiply the Variables with the Same Base

Now, multiply the variables with the same base. Start with the variable \(x\). The exponents of \(x\) are 1 (from \(-\frac{1}{2}xy\)) and 2 (from \(\frac{1}{3}x^2y^3\)). Add these exponents: \(1 + 2 = 3\). So, \(x^3\).

Step 3: Continue with Remaining Variables

Next, handle the variable \(y\). The exponents of \(y\) are 1 (from \(-\frac{1}{2}xy\)) and 3 (from \(\frac{1}{3}x^2y^3\)). Add these exponents: \(1 + 3 = 4\). So, \(y^4\).

Step 4: Combine the Results

Combine the results from the previous steps: the coefficient and the variables. The product is \(-\frac{1}{6}x^3y^4\).

Key Concepts

Exponents

Understanding exponents is crucial in polynomial multiplication. An exponent is a small number placed above and to the right of a base number indicating how many times the base is used as a factor. For instance, in the expression \(x^3\), "3" is the exponent, meaning \(x\) is multiplied by itself three times (\(x \times x \times x\)).
When multiplying variables with exponents, remember to follow this simple rule: **add the exponents if the bases are the same**. This means \(x^a \times x^b = x^{a+b}\).
This rule is applied in polynomial multiplication when combining like terms, helping simplify expressions efficiently.

Variables with the Same Base

In algebraic expressions, you'll often encounter variables with the same base during multiplication. The base is the letter part of the variable, like "x" or "y" in expressions with exponents. When multiplying, only add the exponents if these variables share a base.
For example, in the expression \((xy)(x^2y^3)\), both terms have the base variable \(x\) and \(y\).

• For \(x\), add the exponents: \(x^1\) from the first term and \(x^2\) from the second term become \(x^{1+2} = x^3\).
• For \(y\), add the exponents: \(y^1\) and \(y^3\) become \(y^{1+3} = y^4\).

Understanding how to properly handle variables with the same base is key to simplifying algebraic expressions correctly.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition or multiplication). They are the building blocks of algebra and help us describe mathematical relationships. For instance, the expression \(-\frac{1}{2}xy\) combines a coefficient, variables \(x\) and \(y\), and multiplication.
Expressions become more complex with added variables or operations involved, but the underlying principles remain similar. The expression \(\left(-\frac{1}{2}xy\right)\left(\frac{1}{3}x^2y^3\right)\) is an algebraic expression because it involves multiplying two separate terms, each with its own coefficients and variables.
Understanding how to manipulate these expressions through operations like multiplication allows you to tackle more complex algebraic problems with confidence.

Coefficients

Coefficients are the numerical factors in algebraic expressions that multiply the variables. In an expression like \(-\frac{1}{2}xy\), \(-\frac{1}{2}\) is the coefficient. Coefficients can be positive or negative fractions, integers, or decimals.
To multiply them, simply perform normal arithmetic operations. For the expression \((-\frac{1}{2})(\frac{1}{3})\), multiply the coefficients to get \(-\frac{1}{6}\). This aligns with the first step in multiplying polynomials, ensuring the calculation reflects both the magnitude (absolute value) and direction (positive or negative) of the term.
When working with more complex expressions, remember that coefficients affect the overall value of the product and are essential for calculating the final result correctly.