Problem 42 - Prealgebra

Question

Find each product. Write in simplest form. $$\frac{n}{18} \cdot \frac{6}{n^{4}}$$

Step-by-Step Solution

Answer

\(\frac{1}{3n^3}\) is the simplified product.

Step 1: Multiply the Numerators

To find the product of two fractions, first multiply the numerators together. The numerators are \(n\) and 6. Therefore: \(n \times 6 = 6n\).

Step 2: Multiply the Denominators

Next, multiply the denominators together. The denominators are 18 and \(n^4\). Therefore: \(18 \times n^4 = 18n^4\).

Step 3: Write the Fraction

Combine the results from Steps 1 and 2 to form a single fraction: \(\frac{6n}{18n^4}\).

Step 4: Simplify the Fraction

Now, simplify the fraction. First, factor out the greatest common divisor of the numerical coefficients, which is 6: \(\frac{6n}{18n^4} = \frac{6 \cdot n}{6 \cdot 3 \cdot n^4}\). This simplifies to: \(\frac{n}{3n^4}\).

Step 5: Simplify the Variable Part

Simplify by canceling a factor of \(n\) from both the numerator and denominator: \(\frac{n}{3n^4} = \frac{1}{3n^3}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial skill in mathematics, especially when dealing with multiplication of fractions. When you simplify a fraction, you reduce it to its simplest form - meaning the numerator and the denominator share no common factors except for 1.
To do this, identify the greatest common divisor (GCD) of both the numerator and the denominator and divide them by this number. This process makes the fraction easier to understand and work with in future calculations:

• Factor out the common numbers or variables in both parts of the fraction.
• Cancel out these common factors to simplify the expression.

For example, when simplifying the fraction \(\frac{6n}{18n^4}\), we recognize that both 6 and 18 share a common factor of 6. By removing this factor, the expression becomes simpler. Simplifying doesn’t change the value of the fraction but makes it much easier to handle.

Numerator and Denominator

Every fraction consists of two main components: the numerator and the denominator. Understanding these terms is essential.
The numerator is the top number of a fraction, representing how many parts of a whole are being considered. In contrast, the denominator is the bottom number, indicating how many equal parts the whole is divided into.

• In the fraction \(\frac{n}{18}\), \(n\) is the numerator and 18 is the denominator.
• Similarly, for \(\frac{6}{n^4}\), 6 is the numerator, and \(n^4\) is the denominator.

The process of multiplying fractions involves multiplying the numerators together to get a new numerator and the denominators together to form a new denominator, as can be seen in the expression \(\frac{6n}{18n^4}\). Understanding the roles of both parts helps in performing these operations accurately.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations. When multiplying fractions with algebraic expressions, you need to treat variable components just as you would numerical ones.
Variables in expressions like \(\frac{n}{18} \cdot \frac{6}{n^4}\) are essential for performing operations algebraically, just as numbers do:

• Combine numerical coefficients by multiplying.
• Apply the laws of exponents when dealing with variables. For instance, multiplying \(n\) by \(n^4\) results in \(n^{5}\).
• Simplify by using like terms and reduce the expression to the simplest form if possible.

These steps ensure that your results are both accurate and neatly expressed, essential for solving exercises effectively.