Problem 4 - A Graphical Approach to College Algebra

Question

Each expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$-\frac{2 x^{5}}{7 x^{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{2}{7}x^3\)."

Step 1: Split the Fraction Numerator

According to the given property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\), you can apply similar reasoning to any single-term fraction. However, in this problem, we have the fraction \(-\frac{2x^5}{7x^2}\) where we're only working with a single fraction. Thus, the property won't be directly used, but we'll simplify the expression.

Step 2: Divide Coefficients and Variables

Take the coefficient part and the variable part of the fraction and simplify separately:- Coefficient: \(-\frac{2}{7}\).- Variable: \(\frac{x^5}{x^2} = x^{5-2}=x^3\).

Step 3: Combine the Simplified Results

Combine the simplified results from coefficients and variables:- The simplified coefficient is \(-\frac{2}{7}\).- The simplified variable expression is \(x^3\).Thus, the expression becomes \(-\frac{2}{7}x^3\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and arithmetic operations. Variables usually represent unknown values that can change or vary, and they are represented by letters like \(x\) or \(y\).
When working with algebraic expressions, you can perform operations such as addition, subtraction, multiplication, and division on both constants and variables. Simplifying these expressions can make them easier to understand and solve.
In the original exercise, we started with the expression \(-\frac{2x^5}{7x^2}\). Our goal is to simplify it by following basic algebraic manipulation rules, which leads us to separate the coefficients from the variables. The process shows how combining terms can make expressions much less complicated. Remember to handle each part of the expression carefully to avoid errors.

Exponent Rules

Exponent rules are essential for simplifying expressions involving powers of variables or numbers. They help us manipulate exponents in a way that makes complex expressions more manageable.
One important rule is the division rule, which states that when dividing like bases, you subtract the exponent of the denominator from the exponent of the numerator. Mathematically, it is expressed as \(x^a / x^b = x^{a-b}\).
This rule was applied in the original step where \(x^5 / x^2\) became \(x^{5-2} = x^3\). These rules allow us to simplify expressions efficiently and are fundamental in algebra, especially when dealing with polynomials or any expression with exponential terms.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator is an algebraic expression. Just like numerical fractions, rational expressions can be simplified using similar rules.
In simplifying rational expressions, it is crucial to divide both coefficients and the variables separately. For instance, in the solution, the original fraction \(-\frac{2x^5}{7x^2}\) is split into two parts: the coefficients \(-\frac{2}{7}\) and the variables \(x^5 / x^2\).
This separation helps in simplifying more complex rational expressions, enabling us to express the answer in its simplest form. Rational expressions often arise in algebra, calculus, and various applied mathematical fields, making the understanding of their simplification process very important.