Problem 10
Question
Evaluate each expression. $$4 ! \cdot 5$$
Step-by-Step Solution
Verified Answer
The expression evaluates to 120.
1Step 1: Understand the Factorial
The expression contains a factorial symbol (!). The factorial of a number, in this case 4!, means we multiply 4 by all preceding positive integers. Thus, 4! is equivalent to \(4 \times 3 \times 2 \times 1\).
2Step 2: Calculate the Factorial
Evaluate \(4!\) by performing the multiplication: \(4 \times 3 = 12\), then \(12 \times 2 = 24\), and finally \(24 \times 1 = 24\). So, \(4! = 24\).
3Step 3: Complete the Expression Multiplication
Now that we know \(4! = 24\), we substitute it back into the original expression \(24 \cdot 5\).
4Step 4: Evaluate the Final Multiplication
Multiply 24 by 5 to complete the expression: \(24 \cdot 5 = 120\).
Key Concepts
MultiplicationMathematical ExpressionsOrder of Operations
Multiplication
Multiplication is one of the fundamental operations in arithmetic, alongside addition, subtraction, and division. It involves combining equal groups into a total amount. For example, when we multiply 3 by 4, we are essentially adding 3 groups of 4: \(3 \times 4\) equals 12. This operation can seem straightforward, but it is powerful and widely used in various mathematical contexts.
When we multiply larger numbers or involve complex expressions, it is important to follow a systematic approach. Breaking down the multiplication into smaller parts, as seen in solving factorials, helps reduce errors and simplifies computation. Multiplication is also commutative, which means the order of the numbers can be switched without affecting the outcome: \(a \times b = b \times a\). This property can be very useful, allowing flexibility in solving problems.
Moreover, multiplication plays a crucial role when dealing with sequences and series, as in factorials, where it systematically reduces a sequence of descending positive integers into a product. Understanding how to handle multiplication properly is essential for handling more advanced mathematical tasks.
When we multiply larger numbers or involve complex expressions, it is important to follow a systematic approach. Breaking down the multiplication into smaller parts, as seen in solving factorials, helps reduce errors and simplifies computation. Multiplication is also commutative, which means the order of the numbers can be switched without affecting the outcome: \(a \times b = b \times a\). This property can be very useful, allowing flexibility in solving problems.
Moreover, multiplication plays a crucial role when dealing with sequences and series, as in factorials, where it systematically reduces a sequence of descending positive integers into a product. Understanding how to handle multiplication properly is essential for handling more advanced mathematical tasks.
Mathematical Expressions
Mathematical expressions are combinations of numbers, operators (like \(+\), \(-\), \(\times\), and \(\div\)), and sometimes variables, which stand in for numbers. They can sometimes include symbols like factorials \(!\), which change how the numbers are operated on. Learning how to interpret these expressions is vital in solving mathematical problems.
A factorial, in particular, is a great example of how expressions can encapsulate a lot of information in a compact form. The symbol \(!\) following a number tells you to multiply that number by all smaller positive integers. So, \(4!\) becomes \(4 \times 3 \times 2 \times 1\), which simplifies to 24 as part of evaluating the example expression \(4! \cdot 5\).
Expressions need to be read and solved according to the specific operations they include. They might not always involve just numbers directly but can also involve complexities like parenthesis or order operations that dictate precisely how calculations should be approached to avoid making errors.
A factorial, in particular, is a great example of how expressions can encapsulate a lot of information in a compact form. The symbol \(!\) following a number tells you to multiply that number by all smaller positive integers. So, \(4!\) becomes \(4 \times 3 \times 2 \times 1\), which simplifies to 24 as part of evaluating the example expression \(4! \cdot 5\).
Expressions need to be read and solved according to the specific operations they include. They might not always involve just numbers directly but can also involve complexities like parenthesis or order operations that dictate precisely how calculations should be approached to avoid making errors.
Order of Operations
The order of operations is a fundamental concept in mathematics that dictates the sequence in which operations should be performed to correctly evaluate expressions. In mathematics, this order is commonly memorized through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
Using the correct order of operations ensures that expressions are solved logically and accurately. For instance, in a problem involving multiple operations, such as \(3 + 5 \times 2\), multiplication should be performed first, followed by addition. Thus, \(5 \times 2 = 10\) is done before continuing with \(3 + 10 = 13\).
Applying this concept to our factorial expression \(4! \cdot 5\), after evaluating the factorial \(4!\), we follow with multiplication because there are no parentheses or exponents altering the first operation. This systematic approach prevents errors and ensures the integrity of computations, especially as mathematical expressions become more complex.
Using the correct order of operations ensures that expressions are solved logically and accurately. For instance, in a problem involving multiple operations, such as \(3 + 5 \times 2\), multiplication should be performed first, followed by addition. Thus, \(5 \times 2 = 10\) is done before continuing with \(3 + 10 = 13\).
Applying this concept to our factorial expression \(4! \cdot 5\), after evaluating the factorial \(4!\), we follow with multiplication because there are no parentheses or exponents altering the first operation. This systematic approach prevents errors and ensures the integrity of computations, especially as mathematical expressions become more complex.
Other exercises in this chapter
Problem 10
Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=\frac{n^{2}-1}{n^{2}+1}$$
View solution Problem 10
Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{c}12 \\\12\end{array}\
View solution Problem 11
Write the first five terms of each arithmetic sequence. Do not use a calculator. $$a_{3}=10, d=-2$$
View solution Problem 11
Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$-4,-12,-36,-108, \dots$$
View solution