Problem 29 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{-2 w^{2} x^{3} y}{6 w x^{5} y^{2}}$$

Step-by-Step Solution

Answer

\frac{-1}{3x^{2}y}

Step 1: Identify common factors in the numerator and the denominator

Given the rational expression \(\frac{-2 w^{2} x^{3} y}{6 w x^{5} y^{2}}\), identify common factors in both the numerator and the denominator. The common factors are \(w\), \(x^{3}\), and \(y\).

Step 2: Simplify the numbers

Simplify the numerical coefficients in the expression \(\frac{-2}{6}\). Dividing both by their greatest common divisor, which is 2, we get \(-\frac{1}{3}\).

Step 3: Simplify the variables

Reduce the variables \(w^{2}\) and \(w\). Since \(w^{2} - w = w\), we are left with \(\frac{w^{1}}{w^{1}} = 1\). Similarly, for \(x^{3}\) and \(x^{5}\), \(x^{3} - x^{5} = x^{-2} = \frac{1}{x^{2}}\). Finally, for \(y\) and \(y^{2}\), \(y - y^{2} = y^{-1} = \frac{1}{y}\).

Step 4: Combine the simplified terms

Combine the simplified terms to obtain the reduced expression: \(\frac{-1 \times 1 \times 1}{3 \times x^{2} \times y} = \frac{-1}{3x^{2} y}\).

Key Concepts

lowest terms

When we talk about simplifying a rational expression to its lowest terms, we're aiming to make it as simple as possible. Just like fractions, we want the numerator (top part) and the denominator (bottom part) to have no common factors except 1.

This means that if there is any factor that both the numerator and the denominator share, we should divide both by that factor. For instance, when simplifying \(\frac{-2 w^{2} x^{3} y}{6 w x^{5} y^{2}}\), we look for common factors first.

By identifying and removing common factors, we ensure that our expression is in its simplest form. This helps in making the expression easier to work with, understand, and compare with other expressions. Using the step-by-step approach ensures no step is missed.

common factors

Common factors play a crucial role when simplifying rational expressions. They refer to numbers or variables that both the numerator and the denominator share. In our given expression \(\frac{-2 w^{2} x^{3} y}{6 w x^{5} y^{2}}\), we start by identifying common factors.

Here, the common factors are \(w\), \(x^{3}\), and \(y\). These appear in both the numerator and the denominator at least once. By canceling out these common factors, we're reducing our expression.

To cancel these factors, we subtract the powers of common variables and simplify the numerical coefficients. So for \(w^{2}\) vs. \(w\), we get \(\frac{w}{w} = 1\). Similarly, with \(x^{3}\) and \(x^{5}\), we are left with \(\frac{1}{x^{2}}\). Finally, for \(y\) and \(y^{2} \), it reduces to \(\frac{1}{y} \). This step-by-step method ensures accurate simplification.

numerical coefficients

Numerical coefficients are the numerical parts of the terms in our expression. In \(\frac{-2 w^{2} x^{3} y}{6 w x^{5} y^{2}}\), the numerical coefficients are -2 in the numerator and 6 in the denominator.

To simplify these, we find their greatest common divisor (GCD). The GCD of 2 and 6 is 2. Dividing both coefficients by 2, we get \(\frac{-2}{6} = -\frac{1}{3}\).

It's essential to simplify numerical coefficients first before dealing with variables. This ensures the rational expression is simplified step by step. Once the numerical coefficients are simplified, we focus on the variables and their exponents to complete the reduction process. This systematic approach helps avoid mistakes and makes the expression easier to handle.