Problem 33 - Prealgebra and Introductory Algebra

Question

Multiply. $$\frac{8 x^{2}}{9 y^{3}} \cdot \frac{3 y^{2}}{4 x^{3}}$$

Step-by-Step Solution

Answer

\( \frac{2}{3xy}\)

Step 1: Multiply the fractions

Multiply the numerators together and the denominators together: \(\frac{8 x^{2} \cdot 3 y^{2}}{9 y^{3} \cdot 4 x^{3}}\).

Step 2: Simplify the resulting fraction

The resulting fraction can be simplified as: \(\frac{24 x^{2} y^{2}}{36 x^{3} y^{3}}\) which simplifies further to: \( \frac{2}{3} \frac{x^{2} y^{2}}{x^{3} y^{3}}\).

Step 3: Apply the laws of exponents

The laws of exponents state: \(x^{a} / x^{b} = x^{a - b}\). Apply this here to get: \( \frac{2}{3} x^{2 - 3} y^{2 - 3}\).

Step 4: Calculate final answer

By calculation, \(x^{2 - 3} = x^{-1}\) and \(y^{2 - 3} = y^{-1}\). So the final answer is \( \frac{2}{3} x^{-1} y^{-1}\). This is the same as \( \frac{2}{3xy}\).

Key Concepts

Simplify Fractions

When dealing with fractions, especially in algebra, simplifying them makes the numbers and variables easier to work with. Simplification involves reducing a fraction to its lowest terms. We achieve this by finding common factors in the numerator and denominator and cancelling them out. In our given problem, ewline \[\frac{24x^2y^2}{36x^3y^3}\] was obtained after multiplying. To simplify, we find the greatest common divisor for the numerical values, in our case, 12, and divide both the numerator and denominator by it. Then, we apply the same principle to the variables. Cancel out common variables by subtracting the exponents of like bases in the numerator and denominator. This process leaves us with \[\frac{2}{3}x^{-1}y^{-1}\], a much simpler expression, which further simplifies to \[\frac{2}{3xy}\] when negative exponents are interpreted as reciprocals.

Laws of Exponents

The laws of exponents are critical rules helping us simplify expressions with powers. In the context of our problem, we mainly use the quotient of powers rule, which states that \(a^m / a^n = a^{m - n}\), where 'a' is a base and 'm' and 'n' are exponents. This rule tells us that when dividing like bases, we subtract the exponent in the denominator from the exponent in the numerator.
ewline Applying this rule, we simplify an algebraic fraction by reducing the powers of the variables. For instance, from our step-by-step solution, we apply the law to both 'x' and 'y' variables as follows: \(x^{2 - 3}\) becomes \(x^{-1}\) and \(y^{2 - 3}\) becomes \(y^{-1}\). It is essential to be comfortable with these laws to navigate algebraic manipulation successfully.

Algebraic Fractions

Algebraic fractions are fractions that contain variables in their numerators, denominators, or both. Like numerical fractions, they follow the same rules for operations such as addition, subtraction, multiplication, and division. However, simplifying algebraic fractions often requires a few more steps, as seen when multiplying fractions with variables.
ewline After multiplying the numerators and denominators of the fractions given in the exercise, we must factor and cancel out any common terms according to the laws of exponents and simplification techniques. Being familiar with algebraic fractions is crucial because they often appear in equations and more complex algebraic expressions. By mastering this concept, one can simplify, solve, and manipulate algebraic equations and inequalities much more efficiently.