Problem 15 - Algebra for College Students

Question

Find each product. $$\left(\frac{2}{5} x y^{2}\right)\left(\frac{3}{4} x^{2} y^{4}\right)$$

Step-by-Step Solution

Answer

The product is \(\frac{3}{10} x^3 y^6\).

Step 1: Identify and Multiply Coefficients

The given expression is \(\left(\frac{2}{5} x y^2\right)\left(\frac{3}{4} x^2 y^4\right)\). First, identify the coefficients: \(\frac{2}{5}\) and \(\frac{3}{4}\). Multiply these coefficients together: \(\frac{2}{5} \times \frac{3}{4} = \frac{6}{20}\). Simplify this expression: \(\frac{6}{20} = \frac{3}{10}\).

Step 2: Multiply the Variables with Like Bases

After handling the coefficients, multiply the variables with the same bases. Start with \(x\): the exponents are \(1\) in the first term and \(2\) in the second term. \(x^1 \times x^2 = x^{1+2} = x^3\).

Step 3: Multiply the Variables with Like Bases (Continued)

Continue with \(y\): the exponents are \(2\) in the first term and \(4\) in the second term. \(y^2 \times y^4 = y^{2+4} = y^6\).

Step 4: Combine the Results

Combine the simplified coefficient with the products of the variables: \(\frac{3}{10} x^3 y^6\). This is the final product of the given expression.

Key Concepts

Polynomial Multiplication

In algebra, polynomial multiplication is a fundamental operation that involves multiplying expressions containing variables. A polynomial is an algebraic expression composed of variables and coefficients, combined using addition or subtraction. To multiply polynomials, each term in the first polynomial is multiplied by every term in the second polynomial.

The given exercise involves multiplying two monomials: \(\left(\frac{2}{5}xy^2\right)\) and \(\left(\frac{3}{4}x^2y^4\right)\). Start by multiplying the coefficients, which are the numerical parts of the terms. Then, move on to multiplying the variables while paying attention to their exponents.

When multiplying variables, you combine them by adding the exponents of the same base. This method ensures the result is a single polynomial expression, which is often simpler than the original components.

Coefficients

Coefficients play a critical role in expressing polynomial terms. They are the numerical constants that multiply variables, and they represent the magnitude of the term in the polynomial. For example, in the term \(\frac{2}{5}xy^2\), \(\frac{2}{5}\) is the coefficient.

In the multiplication of polynomials, begin by multiplying the coefficients of each term. For instance, in our example, the coefficients \(\frac{2}{5}\) and \(\frac{3}{4}\) are multiplied, resulting in \(\frac{6}{20}\), which simplifies to \(\frac{3}{10}\).

Simplification of coefficients not only makes the polynomial easier to understand but also provides a cleaner, consolidated final expression. When dealing with fractions as coefficients, simplifying them is an essential step.

Exponents

Exponents indicate how many times a base is multiplied by itself. They are displayed as small numbers placed above and to the right of a base number or variable. In algebraic expressions, understanding exponents is vital because they dictate how terms are combined.

When multiplying expressions with the same base, such as the variables in \(\left(\frac{2}{5}xy^2\right)\left(\frac{3}{4}x^2y^4\right)\), add the exponents. For the variable \(x\), you have \(x^1\) and \(x^2\), which combine to \(x^{1+2}=x^3\). Similarly, for \(y\), \(y^2\) and \(y^4\) combine to \(y^{2+4}=y^6\).

• Exponents simplify repeated multiplication.
• Rules of exponents make polynomial multiplication systematic.
• Adding exponents applies only to terms with the same base.

Recognizing how exponents are manipulated is crucial for effectively simplifying expression products in polynomials.