Problem 5 - Fundamentals of Mathematics

Question

Determine the value of each expression. \(1^{5}\)

Step-by-Step Solution

Answer

The value of the expression \(1^5\) is 1.

Step 1: Understanding the Problem

We need to find the value of the expression \(1^5\). This means we are raising the number 1 to the power of 5.

Step 2: Recalling Rules of Exponents

According to the rules of exponents, any non-zero number raised to an exponent is the number multiplied by itself as many times as the value of the exponent.

Step 3: Calculating the Expression

Since \(1^5\) means multiplying 1 by itself 5 times, we can write it as \(1 \times 1 \times 1 \times 1 \times 1\).

Step 4: Multiplying the Numbers

Multiplying 1 by itself any number of times always results in 1, because 1 is the multiplicative identity. So, \(1^5 = 1\).

Key Concepts

Exponentiation Rules

Exponentiation is a mathematical operation that involves raising a base number to a certain power, known as the exponent. When you see an expression like \(1^5\), it means the number 1 is used as the base and it is raised to the fifth power. The exponent tells us how many times we multiply the base by itself. Here are some basic rules of exponentiation that can help in understanding:

• Any non-zero number raised to the power of 0 is 1. For example, \(3^0 = 1\).
• Any number raised to the power of 1 is itself. For example, \(7^1 = 7\).
• The rule \(a^m imes a^n = a^{m+n}\) applies when multiplying terms with the same base.
• The rule \((a^m)^n = a^{m imes n}\) applies when raising a power to another power.

In the case of the expression \(1^5\), we repeatedly multiply 1 by itself. According to the rules, this results in 1. Understanding these rules can make calculations with exponents much simpler.

Multiplicative Identity

The concept of multiplicative identity is fundamental in arithmetic and algebra. The number 1 is called the multiplicative identity because multiplying any number by 1 leaves it unchanged. This means that regardless of how many times you multiply 1 by itself, the result will always be 1. For example:

• \(1 \times 2 = 2\)
• \(1 \times 3.5 = 3.5\)
• \(1 \times 100 = 100\)

Now, applying this identity to the expression \(1^5\), multiplying 1 by itself five times will still give us 1. It's a simple yet powerful concept because it confirms that certain expressions involving the number 1 are straightforward to evaluate, like \(1^n\), which equals 1 for any integer \(n\).

Mathematics Problem-Solving

Problem-solving in mathematics often involves breaking down a problem into simpler steps, as demonstrated in the calculation of \(1^5\). Here’s how you can approach similar problems effectively:1. **Understand the Problem**: First, clearly identify what is being asked. In the case of \(1^5\), we needed to determine what the expression evaluates to.2. **Recall Relevant Concepts**: Consider the mathematical rules or properties that apply. For this problem, understanding exponents and the multiplicative identity was essential.3. **Perform Calculations**: Solve the problem step by step. Sometimes writing out the process, as with \(1 \times 1 \times 1 \times 1 \times 1\), helps to visualize and ensure accuracy.4. **Verify Your Solution**: Finally, it’s helpful to double-check your work. With simple expressions as \(1^5\), revisiting the rules can confirm the outcome is indeed 1.Applying these general steps to a variety of mathematical problems can aid in building confidence and proficiency. Remember, practice makes perfect and tackling problems systematically will enhance your problem-solving skills over time.