Problem 68 - Elementary and Intermediate Algebra

Question

Perform each division. $$ \frac{24 n^{12}}{8 n^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \(3n^{8}\).

Step 1: Understand the Problem

We're given the expression \( \frac{24n^{12}}{8n^{4}} \). This is a division of two algebraic expressions where both the numerator and the denominator are monomials. Our task is to simplify this expression by performing the division.

Step 2: Divide the Coefficients

First, divide the numerical coefficients. The coefficient in the numerator is 24, and in the denominator, it is 8. Divide 24 by 8 to get 3.

Step 3: Divide the Variables

Next, apply the rules of exponent division to the variables. According to the rule \( \frac{n^{a}}{n^{b}} = n^{a-b} \), where \(a\) and \(b\) are exponents, subtract the exponent of \(n\) in the denominator from the exponent of \(n\) in the numerator: \(12 - 4 = 8\). Thus, \( \frac{n^{12}}{n^{4}} = n^{8} \).

Step 4: Combine the Results

Now, combine the results from Step 2 and Step 3. Multiply the simplified coefficient by the simplified variable expression: \(3 \cdot n^{8} = 3n^{8}\).

Key Concepts

Monomials

In algebra, a **monomial** is a type of expression that consists of only one term. It can include numbers, variables, and their exponents, but all must be part of a single linked sequence by multiplication. For example, in the expression \(24n^{12}\), both 24 and \(n^{12}\) multiply to form one monomial term.

Understanding monomials is crucial because:

• They serve as building blocks for polynomials, which are more complex algebraic expressions.
• The simplicity of monomials helps ease into concepts of arithmetic operations involving algebra.
• Operations like addition, subtraction, multiplication, and division maintain the monomial structure.

When dealing with monomials such as in the expression \(\frac{24n^{12}}{8n^{4}}\), simplifying the expression involves managing both the numerical and variable components meticulously.

Division of Exponents

A fundamental rule in algebra, especially with monomials, is the division of exponents. When you divide terms that have the same base, you subtract the exponents. This concept is guided by the rule:

• \(\frac{n^{a}}{n^{b}} = n^{a-b}\), where 'n' is the base and 'a' and 'b' are the exponents.

Applying this to our example \(\frac{24n^{12}}{8n^{4}}\), we take the exponents from both the numerator and the denominator for 'n':

• Numerator: \(n^{12}\)
• Denominator: \(n^{4}\)

Subtract the exponent 4 from 12 (i.e., \(12 - 4\)) to get \(n^{8}\). This operation simplifies the variable part of the expression significantly.

Remember, division of exponents applies only to variables sharing the same base; otherwise, other rules may be necessary.

Simplifying Expressions

The goal of simplifying algebraic expressions is to rewrite them into their simplest form. This means reducing the expression to the fewest possible terms without changing its value. The steps typically include handling coefficients and applying rules like those for exponents carefully.

In our example \(\frac{24n^{12}}{8n^{4}}\), simplifying involves these steps:

• Start by simplifying numerical coefficients: \(\frac{24}{8} = 3\).
• Next, simplify the variables using division of exponents: \(n^{8}\).
• Combine results: \(3 \cdot n^{8} = 3n^{8}\).

These actions turn a complex-looking expression into a more manageable form. Simplifying expressions is an essential algebra skill, helping students reason logically and solve more advanced math problems efficiently.