Problem 46 - Beginning and Intermediate Algebra

Question

Evaluate. $$ (-1)^{5} $$

Step-by-Step Solution

Answer

The result of \((-1)^{5}\) is \(-1\).

Step 1: Understand the Expression

The expression given is \((-1)^{5}\). This means that we have raised the base \(-1\) to the power of 5.

Step 2: Understand the Concept of Exponents

An exponent denotes repeated multiplication. Thus, \((-1)^{5}\) translates to multiplying five instances of \(-1\) together: \((-1) \times (-1) \times (-1) \times (-1) \times (-1)\).

Step 3: Multiply in Pairs

Let's multiply the first pair: \((-1) \times (-1) = 1\). Repeat for the next pair: \((-1) \times (-1) = 1\). Now we have the expression \(1 \times 1 \times (-1)\).

Step 4: Final Multiplication

Multiply the remaining numbers: \(1 \times 1 = 1\), and finally, \(1 \times (-1) = -1\).

Step 5: Conclusion

After evaluating the multiplication, the final result of \((-1)^{5}\) is \(-1\).

Key Concepts

Multiplication

Multiplication is a basic arithmetic operation, but it plays a significant role in understanding exponents. Here’s a simple breakdown to help you grasp the concept better.Think of multiplication as repeated addition. For example, when you multiply 2 by 3, it is the same as adding 2 three times: 2 + 2 + 2 = 6. When it comes to exponents, multiplication serves as the foundation, because raising a number to a power indicates you should multiply the base by itself multiple times.To tackle the expression \((-1)^{5}\), start by understanding that it signifies five instances of multiplying \(-1\) with itself. Sequentially multiplying these numbers involves understanding the product of pairs or groups, breaking down complex multiplications into more manageable steps.Some simple tips to remember:

• Any number times 1 remains unchanged (e.g., 7 \(\times\) 1 = 7).
• Multiplying two negative numbers results in a positive product (e.g., \(-2 \times -3 = 6\)).
• Multiplying a negative number by a positive number results in a negative product (e.g., \(-5 \times 2 = -10\)).

Understanding these rules helps simplify problems involving multiplication, especially when dealing with negative numbers and exponents.

Negative Numbers

Negative numbers can sometimes seem tricky, especially if the rules are not clear. But with a few basic principles, you can master their multiplication and use in exponents with ease.In the expression \((-1)^{5}\), it is important to understand what happens when you multiply negative numbers together:

• Multiplying two negative numbers results in a positive number. For example, \(-1 \times -1 = 1\).
• Multiplying a positive number by a negative number yields a negative result, such as \(1 \times -1 = -1\).
• When multiplying an odd number of negative numbers, the result will be negative. In \((-1)^{5}\), you're multiplying it an odd number of times, leading to a negative outcome.

Understanding these key rules allows you to predict the outcome when multiplying groups of negative numbers. This is crucial when you encounter negative bases raised to any power, helping you determine whether the result will be positive or negative.

Power Rule

The power rule is an essential concept in mathematics that makes dealing with exponents straightforward and systematic. It’s essentially about multiplying a number by itself a certain number of times, dictated by the exponent.For example, given \((-1)^{5}\), consider the rule that tells us how to work with powers. When a base is raised to a power, it means you multiply the base by itself as many times as the exponent indicates. The general guidelines for using the power rule include:

• An exponent of 0 will always yield 1 (e.g., \(a^{0} = 1\) provided \(a\) is not zero).
• An exponent of 1 is the number itself (e.g., \(a^{1} = a\)).
• For negative numbers with odd exponents, the result remains negative (as demonstrated, \((-1)^{5} = -1\)).
• Negative numbers with even exponents yield positive results, such as \((-1)^{4} = 1\).

This power rule helps determine the nature of the outcome without manually having to multiply each step, which is particularly useful for larger exponents and reduces computational complexity.