Problem 23 Divide. Write each answer in low... [FREE SOLUTION] (2024)

Short Answer

Simplifying Fractions

In the world of algebra, simplifying fractions is all about reducing a fraction to its lowest possible terms. This involves finding common factors in the numerator and the denominator and canceling them out. To illustrate, consider the fraction \(\frac{4x}{8y}\). Both the numerator (4x) and the denominator (8y) share a common factor of 4. By dividing both terms by this common factor, we get \(\frac{x}{2y}\). The fraction is now simplified. Simplifying your fractions makes them easier to work with and can reveal more about the behavior of the algebraic expression.

Multiplying Fractions

Multiplying fractions in algebra follows a straightforward rule: multiply the numerators together and the denominators together. Let's revisit our example from the original exercise. When given \(\frac{2x}{x-1} \times \frac{x+2}{x^2}\), multiply the numerators \(2x (x+2)\) to get the new numerator. Then, multiply the denominators \((x-1) x^2\) to get the new denominator. This gives us \(\frac{2x(x+2)}{(x-1) x^2}\). Multiplying fractions is a fundamental skill that makes handling algebraic expressions much simpler, especially when dealing with rational expressions in equations and inequalities.

Algebraic Expressions

Understanding algebraic expressions is critical in solving problems involving algebraic fractions. An algebraic expression is a combination of variables, coefficients, and constants formed using addition, subtraction, multiplication, and division. In the problem \(\frac{2x}{x-1} \times \frac{x+2}{x^2}\), each term—2x, x-1, x+2, and x^2—is an algebraic expression. When these expressions interact with each other through operations like multiplication and division, we must follow algebraic rules to simplify and solve them. A key aspect of working with algebraic expressions is recognizing common factors and canceling them to simplify the fraction. This simplification process helps in making complex algebraic problems more manageable.