Problem 18 - Algebra for College Students

Question

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{4 x^{2}}{5 y^{2}} \cdot \frac{15 x y}{24 x^{2} y^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x}{2y^3}\).

Step 1: Multiply the Numerators and Denominators

First, multiply the numerators together and the denominators together. For the numerators, multiply \(4x^2\) by \(15xy\). For the denominators, multiply \(5y^2\) by \(24x^2y^2\). As a result, you get the fractions \(\frac{4x^2 \cdot 15xy}{5y^2 \cdot 24x^2y^2}\).

Step 2: Expand the Products

Expand the expression from Step 1. The expression \(4x^2 \cdot 15xy\) becomes \(60x^3y\). The expression \(5y^2 \cdot 24x^2y^2\) becomes \(120x^2y^4\). Thus, the expanded expression is \(\frac{60x^3y}{120x^2y^4}\).

Step 3: Simplify the Fraction

In this step, we will simplify the expression by cancelling out common factors from the numerator and the denominator. Divide both the numerator and the denominator by the greatest common factor of 60 and 120, which is 60. Cancel the \(x^2\) in the numerator and the denominator, and \(y\) in the numerator and \(y^4\) in the denominator. You get \(\frac{x}{2y^3}\).

Key Concepts

Simplifying Expressions

When working with mathematical expressions, simplifying them makes solving problems easier and more intuitive. Simplifying involves rewriting an expression in its simplest form while keeping the same value. Rewriting the expression can involve reducing fractions, combining like terms, or removing any common factors.

For example, given a rational expression \(\frac{60x^3y}{120x^2y^4}\), simplifying it involves:

• Identifying and canceling common factors from the numerator and the denominator.
• In this case, you divide the numerator and the denominator by their greatest common factor, which is 60.
• Cancel common terms like \(x^2\) and \(y\), resulting in a simpler expression \(\frac{x}{2y^3}\).

This process not only makes calculations easier but also helps in recognizing equivalent expressions.

Multiplying Fractions

Multiplying fractions is a straightforward process. It involves multiplying across the numerators and denominators. This means that to multiply two or more fractions, you multiply all the numerators to form a new numerator, and all the denominators to form a new denominator.

In our case:

• We have \(\frac{4x^2}{5y^2} \cdot \frac{15xy}{24x^2y^2}\).
• Multiply the numerators, \(4x^2\) and \(15xy\), to get \(60x^3y\).
• Next, multiply the denominators, \(5y^2\) and \(24x^2y^2\), to get \(120x^2y^4\).

Once you have your new fraction, simplify it if possible to make sure the expression is in its most reduced form.

Factoring Polynomials

Factoring polynomials is a key technique used to simplify rational expressions. Factoring means finding the expression that, when multiplied, will give back the original polynomial. This is essential in simplifying rational expressions, as it allows for canceling of common terms.

For instance:

• Consider the expression \(x^3 + 2x^2 + x\). We can factor out the greatest common factor, \(x\), leading to \(x(x^2 + 2x + 1)\).
• This can further be factored if possible to simplify it even further.
• In the context of rational expressions, factoring is used to identify common terms in the numerator and denominator, allowing for cancellations.

Understanding how to effectively factor can make simplifying a rational expression much easier and more intuitive.