Problem 76 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ (-1)^{-1}(-1)^{-1} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \((-1)^{-1}(-1)^{-1}\) with only positive exponents and find its value. Answer: The given expression with only positive exponents is \((-1)^{-1}(-1)^{-1} = 1\).

Step 1: 1. Apply properties of exponents

We need to multiply the given exponential expressions. When multiplying exponential expressions with the same base, we can just add the exponents: $$ (-1)^{-1}(-1)^{-1} = (-1)^{(-1) + (-1) } $$

Step 2: 2. Simplify the expression

Now, sum the exponents and find the simplified expression: $$ (-1)^{-1}(-1)^{-1} = (-1)^{(-1) + (-1)} = (-1)^{-2} $$

Step 3: 3. Convert to positive exponent

To convert the result to a positive exponent, we can use the property \(a^{-n} = \frac{1}{a^n}\): $$ (-1)^{-2} = \frac{1}{(-1)^2} $$

Step 4: 4. Calculate the final result

Finally, evaluate the expression in the denominator: $$ (-1)^{-2} = \frac{1}{(-1)^2} = \frac{1}{1} $$ So, the given expression with only positive exponents is: $$ (-1)^{-1}(-1)^{-1} = 1 $$

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial for simplifying and working with exponential expressions. Exponents, also known as powers, are a shorthand way to express repeated multiplication of a number by itself. When dealing with exponents, there are several rules or properties that can make calculations easier. First, when you multiply numbers with the same base, you add the exponents. For example, \( a^m \cdot a^n = a^{m+n} \). Second, for any nonzero number \( a \), the reciprocal can be expressed as an exponent by \( a^{-n} = \frac{1}{a^n} \). This is particularly helpful when converting negative exponents to positive ones, as seen in the textbook exercise.

In the exercise example, we apply these rules by first adding the exponents of the same base and then flipping the base to get rid of the negative exponent. These properties allow for a streamlined approach to handling complex expressions and play a fundamental role in algebra and calculus.

Simplifying Expressions

Simplifying expressions is like cleaning up a room: it’s about making the expression as neat as possible, without changing its value. When simplifying, we use the properties of exponents to combine like terms, convert negative to positive exponents, or break down complex expressions into something more manageable. But why simplify? Well, it’s not just for aesthetics; simpler expressions are easier to understand and work with, particularly as they become part of larger algebraic equations or calculus problems. In our exercise example, simplification helps us to see that \((-1)^{-1}(-1)^{-1}\) is actually the same as 1, even though it initially looks complex with negative exponents. Simplifying expressions is an essential skill in mathematics, helping students to solve problems effectively and reducing the potential for errors.

Exponential Expressions

Exponential expressions are powerful tools in mathematics, used to represent rapid growth or decay, such as population growth, interest rates, and radioactive decay. An exponential expression is written as \( a^n \), where \( a \) is the base, and \( n \) is the exponent, which tells us how many times the base is multiplied by itself. The exercise example is a basic form of an exponential expression involving negative exponents. When working with exponential expressions, it's essential to remember not only the properties mentioned above but also to pay attention to the base. If the base is negative, like -1, it affects the outcome as it flips the sign each time it's multiplied. Understanding how to manipulate these expressions mathematically to reveal their simplified form is critical in many areas of study. Mastery of exponential expressions is not only important for academic success but is also practical in understanding real-world phenomena.