Problem 16 - Prealgebra

Question

Find each product. Write in simplest form. $$\frac{4 t}{9 r} \cdot \frac{18 r}{t^{2}}$$

Step-by-Step Solution

Answer

\(\frac{8}{t}\) is the simplest form.

Step 1: Write the expression

First, write the given expression in its complete fractional form:\[\frac{4t}{9r} \cdot \frac{18r}{t^2}\]

Step 2: Multiply the numerators

Multiply the numerators of the two fractions. This involves multiplying 4t by 18r. The product is:\[4t \cdot 18r = 72tr\]

Step 3: Multiply the denominators

Multiply the denominators of the two fractions. This involves multiplying 9r by \(t^2\). The product is:\[9r \cdot t^2 = 9rt^2\]

Step 4: Combine into a single fraction

Place the results from Steps 2 and 3 into a single fraction:\[\frac{72tr}{9rt^2}\]

Step 5: Simplify the fraction

To simplify, start by canceling common factors. Both the numerator and the denominator share factors:- Cancel the \(r\) in the numerator and denominator.- Cancel \(t\) in the numerator and one \(t\) from \(t^2\) in the denominator.- Now divide the coefficients 72 and 9 by their greatest common divisor, which is 9.This results in:\[\frac{72tr}{9rt^2} = \frac{8}{t}\]

Step 6: Write the final simplified result

The expression is simplified to:\[\frac{8}{t}\]

Key Concepts

Multiplying Fractions

To multiply fractions, follow this simple two-step process:

• Multiply the numerators: The numerator is the top part of a fraction. When multiplying fractions, simply multiply the numerators together to get the new numerator.
• Multiply the denominators: The denominator is the bottom part of a fraction. Multiply the denominators of both fractions to get the new denominator.

For example, when multiplying \(\frac{4t}{9r}\) with \(\frac{18r}{t^2}\), we multiply:

• Numerators: \(4t \cdot 18r = 72tr\)
• Denominators: \(9r \cdot t^2 = 9rt^2\)

This results in a single fraction \(\frac{72tr}{9rt^2}\). Breaking down each step ensures clear and precise calculations, making the multiplication of fractions much simpler. It's crucial to take this step-by-step approach to avoid confusion.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition, subtraction, multiplication, and division). When dealing with algebraic expressions, each variable and number behaves according to specific rules. Let's consider the variables in our example:

• Variables involved: \(t\) and \(r\). These variables represent unknowns or quantities that can change.
• Product and Quotient Rules: When multiplying variables like \(4t \cdot 18r\), treat each variable separately. Here, you multiply like terms. Similarly, when simplifying \(\frac{72tr}{9rt^2}\), you simplify by canceling matching terms.

Understanding how to manipulate these expressions is key in algebra. With practice, working with algebraic expressions becomes more intuitive, and simplifying them becomes second nature.

Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator and the denominator. They follow the same rules as ordinary fractions but are sometimes more complex due to their algebraic nature. Here are key points to remember when working with rational expressions:

• Cancel Common Factors: To simplify, you must identify and cancel any common factors from the numerator and the denominator. For example, in \(\frac{72tr}{9rt^2}\), you can cancel the common factor \(r\) and one \(t\).
• Simplification of Coefficients: Reduce the coefficients by dividing them by their greatest common divisor (GCD). In the example, 72 and 9 share a GCD of 9, so divide both by 9 to obtain \(8\).
• Final Reduced Form: After canceling all possible common factors and simplifying the coefficients, you get \(\frac{8}{t}\), the simplest form of that rational expression.

Mastering rational expressions involves systematically reducing and simplifying these expressions to their lowest terms. It is a vital skill for progressing in both algebra and higher-level mathematics.