Problem 8 - Elementary Algebra

Question

For Problems \(1-24\), divide the monomials. $$ \frac{84 x^{5}}{-7 x^{5}} $$

Step-by-Step Solution

Answer

The quotient is \(-12\).

Step 1: Identify the Problem

The problem requires finding the quotient when dividing two monomials: \( \frac{84x^5}{-7x^5} \).

Step 2: Divide the Coefficients

Start by dividing the numerical coefficients of the monomials: \( \frac{84}{-7} = -12 \).

Step 3: Divide the Variables

Since the variables \( x^5 \) in the numerator and denominator are the same, they cancel each other out, simplifying the expression to \( x^{5-5}=x^0=1 \).

Step 4: Combine the Results

Combine the results from the numerical division and the variable division: \( -12 \times 1 = -12 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are formed by combining numbers and variables using arithmetic operations like addition, subtraction, multiplication, and division. In our exercise, we deal with a specific type of algebraic expression called a "monomial." A monomial consists of a single term that can include a numerical coefficient and variables raised to non-negative integer exponents. In the division of monomials, we apply operation rules to both the coefficients and the variables. The goal is to simplify the expression by determining the quotient of the given monomials. It is crucial to recognize the components, such as numerical coefficients and variables, correctly, as this understanding forms the foundation for simplifying the expression.

Exponent Rules

Exponent rules are mathematical principles used to simplify the multiplication and division of terms that involve powers. One key rule for division is that when you divide like bases, you subtract the exponents. For instance, in the problem, we have the variable part as \( x^5 \) divided by \( x^5 \). The rule applied here is \( x^m \div x^n = x^{m-n} \). Therefore, \( x^5 \div x^5 = x^{5-5} = x^0 \). Moreover, any number raised to the power of zero is equal to one, which is why \( x^0 = 1 \). This rule helps in simplifying expressions by effectively reducing the power terms, which is essential for simplifying algebraic expressions involving division.

Numerical Coefficients

Numerical coefficients are the constant numbers found in terms of an algebraic expression. They signify the number by which the variable part of the term is multiplied. In the monomial division involving \( \frac{84x^5}{-7x^5} \), the coefficients are 84 and -7. To find the quotient of these coefficients, we perform a simple division: \( \frac{84}{-7} = -12 \). When dividing monomials, evaluating the numerical coefficients is a critical step because it simplifies one component of the expression. Combining this quotient with the result of dividing the variable parts gives us a complete simplified result of the original problem.