Problem 46 - Algebra 2

Question

PREREQUISITE SKILL Evaluate each expression. $$ 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 $$

Step-by-Step Solution

Answer

The value of the expression is 120.

Step 1: Understand the Expression

The expression given is a series of consecutive numbers from 5 down to 1 being multiplied. This is a product of integers and can be viewed as a factorial of 5, denoted by 5!.

Step 2: Calculate the First Part of the Product

Start by multiplying the first two numbers:\[ 5 \cdot 4 = 20 \]

Step 3: Continue with the Next Multiplication

Take the result from Step 2 and multiply it by the next number in the sequence:\[ 20 \cdot 3 = 60 \]

Step 4: Multiply by the Next Number

Use the result of Step 3 to continue the sequence:\[ 60 \cdot 2 = 120 \]

Step 5: Conclude with the Final Multiplication

Finally, multiply the result from Step 4 by the last number in the sequence:\[ 120 \cdot 1 = 120 \]

Step 6: Final Result

The final result of multiplying all the numbers in the sequence is 120.

Key Concepts

Integer Multiplication

Integer multiplication is the process of finding the total by adding a number to itself, a specified number of times. In this context, all numbers involved are whole numbers, meaning they don't have fractions or decimals. When you multiply, you're essentially finding the product of two or more whole numbers. For example, in the problem \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\), each multiplication step adds up the integers in sequence:

• First, multiply 5 by 4 to obtain 20.
• Then, take the result and multiply by the next integer, 3, to get 60.
• Proceed by multiplying 60 by 2, resulting in 120.
• Finally, multiply by 1, but the result remains the same: 120.

Remember, multiplying by 1 doesn't change the number. It's like the identity property of multiplication. Understanding integer multiplication in sequences like this helps in calculating more complex factorials and series.

Consecutive Numbers

Consecutive numbers are numbers that follow each other in order, without any gaps, from smallest to largest. In our expression \(5, 4, 3, 2, 1\), these numbers are placed in a sequence and each number in the sequence is one unit less than the number before it. In mathematics, consecutive integers are often involved in factorials, like the one you encountered. They form the foundation for the calculations.

• The concept of consecutive numbers keeps things in order, ensuring nothing is skipped or repeated, which is crucial for accurate calculations.
• Using consecutive numbers makes predicting patterns or simplifying problems more straightforward, sometimes allowing mental math with practice.

Understanding this concept helps in solving problems involving series or sequences, as it allows you to systematically work through every step.

Product of Integers

The product of integers refers to the result obtained after multiplying two or more integers together. In this context, the product you're finding with \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\) is 120. Each integer multiplied contributes to the final product. A product of integers can increase dramatically with each additional number, especially when dealing with larger numbers or longer sequences. In our case, each step built on the previous one:

• You start with small products then expand by multiplying again, e.g., from 5 and 4 giving 20.
• Then continue multiplying until all integers in the list have been included in the product.

The idea here is to see how multiple numbers together produce one single result, illustrating the force of multiplication and why order and method matter. Understanding these fundamentals aids when tackling more complex equations or factorial problems in the future.