Problem 22 - Elementary and Intermediate Algebra

Question

Simplify. $$ (-1)^{7} $$

Step-by-Step Solution

Answer

The simplified value is \(-1\).

Step 1: Understand the Problem

The problem requires finding the value of \( (-1)^7 \).

Step 2: Recall the Rule for Exponents

Recall that raising \( -1 \) to an odd power always results in \(-1\).

Step 3: Apply the Rule

Since 7 is an odd number, \( (-1)^7 = -1 \).

Key Concepts

Odd Exponents

Exponentiation is one of the fundamental mathematical operations. When you raise a number to an exponent, you are essentially multiplying that number by itself a certain number of times. Specifically, an odd exponent means you multiply the base an odd number of times. For example, raising a number to the power of 7 (7{}) means multiplying the number by itself seven times.
It's worth noting that when the exponent is odd, the sign of the base will heavily influence the final result. If the base number is positive and you raise it to an odd exponent, the result is positive. However, if the base number is negative and the exponent is odd, the result remains negative.

Negative Base

A negative base changes the rules a bit when it comes to exponentiation. If you have a negative base and raise it to an even exponent, the result will be positive. But if you raise a negative base to an odd exponent, the result will be negative.

For example, consider (-1)^2 (-1)^2 which results in 1 because multiplying an even number (2) flips the sign back to positive. However, if you have (-1)^3 (-1)^3, it results in (-1)because multiplying an odd number (3) keeps the negative sign.

This general rule will help you understand and simplify expressions involving negative bases and exponents.

Simplification

Simplifying expressions with exponents can seem daunting, but it's all about understanding the rules. With the given exercise, (-1)^7 (-1)^7, we can simplify by noting the exponent is odd.

Since 7 is an odd number and the base -1 is negative, the result follows a specific rule: any negative base raised to an odd number will remain negative. In this case, (-1)^7 = -1 (-1)^7 = -1.

Therefore, the result of (-1)^7 (-1)^7 is -1 -1. Through these steps, you can see that knowing the basic properties of exponents and bases can significantly simplify the process.