Problem 73 - Algebra for College Students

Question

Find each quotient. $$\frac{-36 x^{3} y^{5}}{2 y^{5}}$$

Step-by-Step Solution

Answer

\(-18 x^3\)

Step 1: Simplify the Coefficients

To simplify the coefficients of the expression \(\frac{-36 x^3 y^5}{2 y^5}\), divide -36 by 2. \(-36 \div 2 = -18\). This gives us the new expression: \(-18 x^3 y^5\).

Step 2: Cancel Out the Common Variables

Notice that \(y^5\) appears in both the numerator and the denominator. Cancel \(y^5\) in the numerator with \(y^5\) in the denominator. This leaves us with \(-18 x^3\).

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves breaking down the expression into its simplest form. The goal is to make the expression as straightforward as possible by performing operations such as adding, subtracting, multiplying, or dividing coefficients and terms. In our example, \[\frac{-36 x^{3} y^{5}}{2 y^{5}}\] we start with simplifying the coefficients. The coefficient -36 divided by 2 gives us -18. This operation simplifies the expression significantly, reducing the complexity by focusing only on terms that need further simplification or cancellation. Also, different parts of the expression can consist of coefficients, variables, and exponents. Simplifying bring these parts together into a more concise form, making it easier to understand and work with other algebraic processes.

Dividing Monomials

When dividing monomials, it's essential to understand that the process is similar to dividing regular numbers or fractions. In the context of polynomials, this involves dividing the coefficients and subtracting the exponents of like bases. Take our example where we already simplified the coefficients: \(-18 x^{3} y^{5}\). Here, we understand that dividing a monomial involves handling the terms separately:

• Coefficients: The numbers in front of the variables are simplified by regular division (e.g., -36 divided by 2 becomes -18).
• Variables: For any variable terms such as \(x^3\), division entails subtracting exponents when variables are similar on the numerator and denominator.

Understanding how to correctly divide monomials facilitates the simplification of complex algebraic expressions, turning them into manageable components that reveal the essential character of the expression.

Canceling Common Factors

Canceling common factors is a crucial step in simplifying algebraic expressions. This technique involves identifying and removing the same terms present on both the numerator and the denominator. In our problem, you can observe the presence of \(y^5\) both above and below the fraction line in \[\frac{-36 x^{3} y^{5}}{2 y^{5}}\]. Canceling these terms, essentially a division of the same factors, simplifies the expression to \(-18 x^3\). This effectively reduces the expression to its simplest form by eliminating redundant information and allowing focus on surviving terms that carry distinct value. This not only simplifies calculations but also makes it easier to work with algebraic expressions during further operations or manipulations. When rightly understood, canceling common factors becomes an intuitive instinct that greatly enhances problem-solving skills, particularly in algebra.