Problem 18 - Elementary and Intermediate Algebra

Question

Divide the monomials. See Example 1. $$ \frac{-4 x^{3}}{16 x^{5}} $$

Step-by-Step Solution

Answer

\(-\frac{1}{4x^2}\)

Step 1: Identify Coefficients and Variables

The numerator is \(-4x^3\) and the denominator is \(16x^5\). First, identify the coefficients and variables with exponents in both the numerator and the denominator. Here, the coefficient in the numerator is \(-4\) and in the denominator, it's \(16\). The variable \(x\) has an exponent of 3 in the numerator and 5 in the denominator.

Step 2: Divide the Coefficients

Divide the coefficients \(-4\) by \(16\). This results in \(-\frac{1}{4}\). We divide the coefficients separately without involving the variable part.

Step 3: Subtract Exponents of Like Bases

Since both the numerator and the denominator contain the variable \(x\), subtract the exponent in the denominator from the exponent in the numerator: \(3 - 5 = -2\). This gives us \(x^{-2}\).

Step 4: Simplify the Result

Combine the coefficient with the variable: \(-\frac{1}{4}x^{-2}\). Since negative exponents indicate reciprocal, simplify to \(-\frac{1}{4x^2}\) to have only positive exponents.

Key Concepts

Coefficients and Variables

When dividing monomials, it is important to identify both coefficients and variables in the given terms. To illustrate, let's revisit the exercise where we have two parts: the numerator \(-4x^3\) and the denominator \(16x^5\). - **Coefficients**: These are the numerical factors associated with the variables. In our situation, the coefficient of the numerator is \(-4\), while the denominator's coefficient is \(16\).- **Variables**: These are represented by letters (like \(x\) in our example) which can have exponents. In the numerator, the variable \(x\) has an exponent of \(3\) and in the denominator, it's \(5\).Understanding which parts are coefficients and which are variables is crucial. This distinction enables you to deal with the numerical and algebraic parts separately, streamlining the division process.

Subtracting Exponents

When both the numerator and the denominator feature like bases, you subtract the exponents as part of the division process. Here, the like base in the numerator and denominator is the variable \(x\).Given our example:- The exponent of \(x\) in the numerator is \(3\)- In the denominator, it is \(5\)To divide, subtract the exponent in the denominator from that in the numerator: \(3-5 = -2\).The result is an exponent of \(-2\), giving us \(x^{-2}\). This process of subtracting exponents with the same base is fundamental when handling exponents during division. This approach consistently sorts out the power relationship between like bases.

Negative Exponents

Handling negative exponents properly is essential in algebra. A negative exponent suggests a reciprocal action. For instance, in our solved problem, \(x^{-2}\) surfaced after subtracting exponents.- **Reciprocal Rule**: When you encounter a negative exponent, rewrite it as a reciprocal. Thus, \(x^{-2}\) converts to \(\frac{1}{x^2}\).Simplifying \(-\frac{1}{4}x^{-2}\) by rewriting the negative exponent transforms it into \(-\frac{1}{4x^2}\). This conversion simplifies expressions and aligns with the principle of keeping exponents positive in final answers, which helps in maintaining clarity and simplicity in the results.