Problem 48 - Algebra 2

Question

PREREQUISITE SKILL Evaluate each expression. $$ \frac{9 \cdot 8 \cdot 7 \cdot 6}{4 \cdot 3 \cdot 2 \cdot 1} $$

Step-by-Step Solution

Answer

The value of the expression is 126.

Step 1: Break Down Numerator

The numerator of the fraction is given as \(9 \cdot 8 \cdot 7 \cdot 6\). Start by calculating the product of these numbers. First, multiply 9 and 8: \(9 \times 8 = 72\). Next, multiply 72 by 7: \(72 \times 7 = 504\). Finally, multiply 504 by 6: \(504 \times 6 = 3024\). Thus, the numerator evaluates to 3024.

Step 2: Simplify Denominator

The denominator of the fraction is \(4 \cdot 3 \cdot 2 \cdot 1\). Calculate the product of these numbers. Start by multiplying 4 and 3: \(4 \times 3 = 12\). Then multiply 12 by 2: \(12 \times 2 = 24\). Finally, multiply 24 by 1: \(24 \times 1 = 24\). Therefore, the denominator evaluates to 24.

Step 3: Divide Numerator by Denominator

Now that you have simplified both the numerator and the denominator, divide the numerator by the denominator to evaluate the expression: \(\frac{3024}{24}\). Perform the division to get \(3024 \div 24 = 126\). Therefore, the expression simplifies to 126.

Key Concepts

Understanding Numerator and Denominator

In rational expressions, the fraction is divided into two essential parts: the numerator and the denominator. The **numerator** is the top part of the fraction, representing the number of parts we have. In our example, the numerator is a product of 9, 8, 7, and 6, leading to a numerical value of 3024.

The **denominator** is the bottom part of the fraction, indicating the number of parts the whole is divided into. Here, the denominator is derived from multiplying 4, 3, 2, and 1, resulting in 24.

Understanding these components helps in grasping how a fraction functions, with the numerator needing to be divided by the denominator to find the value of the fraction.

Multiplication in Fractions

Multiplication is a key operation when working with fractions like our example. The process involves multiplying the numbers listed in both the numerator and the denominator.

To multiply, follow these basic steps:

• Multiply the numbers consecutively to handle larger numbers step-by-step.
• This helps in breaking down complex calculations into simple ones.
• As seen in the problem, multiplying 9 by 8 gave 72, further multiplying by 7 gave 504, and then by 6 to yield the numerator of 3024.
• Similarly, multiply 4, 3, 2, and 1 to get the denominator of 24.

Applying these multiplication principles is essential to evaluating fractions accurately.

Division of Numerator by Denominator

After obtaining the products of both the numerator and the denominator through multiplication, the next step is to perform division. Division helps in determining the simplest form or value of the fraction.

Take the final product of the numerator, 3024, and divide it by the final product of the denominator, 24.

• This division enables us to calculate the value of the fraction.
• Performing the calculation, we find that 3024 divided by 24 equals 126.

The outcome of this division provides us with the value of the original expression, effectively simplifying it from a complex fraction to a simple number.

Simplifying Fractions

Simplifying fractions is about reducing them to their simplest form where the numerator and the denominator have no common divisors other than 1. This makes the fraction easier to work with and understand.

In our given expression, simplifying is a process of division to reduce the complexity of the original fraction. By dividing 3024 by 24, we reduced the fraction to its simplest form, resulting in a value of 126.

• Check for common factors if the numbers are not apparent, usually using division.
• When completely simplified, fractions provide clear insights into the number's value.

This step not only makes the fraction easier to handle but also enhances accuracy in mathematical calculations.