Problem 37 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ (-2)^{-1} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression $$(-2)^{-1}$$ using only positive exponents. Answer: $$\frac{1}{-2}$$

Step 1: Understand the negative exponent rule

The rule for negative exponents states that $$a^{-n} = \frac{1}{a^n}$$. This means that any expression with a negative exponent can be rewritten as the reciprocal of the expression with the positive exponent.

Step 2: Apply the negative exponent rule

Using the negative exponent rule, we can rewrite the given expression as: $$(-2)^{-1} = \frac{1}{(-2)^1}$$

Step 3: Simplify the expression

Lastly, we simplify the expression: $$\frac{1}{(-2)^1} = \frac{1}{-2}$$

Key Concepts

Exponent Rules

When dealing with exponents, it is crucial to understand the various rules that govern their operations, especially negative exponents. The negative exponent rule states that any base raised to a negative exponent means that you take the reciprocal of the base raised to the corresponding positive exponent. For example, when you have a negative exponent like \((-2)^{-1}\), it can be expressed as \(\frac{1}{(-2)^1}\), by applying the negative exponent rule.

• This rule simplifies calculations by removing negative exponents and replacing them with positive ones in the denominator or numerator.
• It is part of a larger set of exponent rules, like the power of a power rule \((a^m)^n = a^{mn}\) and product of powers rule \(a^m \times a^n = a^{m+n}\).

These rules help manage and manipulate expressions easily, ensuring accurate simplification and calculation.

Algebraic Expressions

Algebraic expressions are a combination of variables, numbers, and constants, along with arithmetic operations like addition, subtraction, multiplication, and division. In our specific exercise, we see an algebraic expression involving an exponent: \((-2)^{-1}\).

• Expressions can differ in complexity, from simple constants and variables to multi-term polynomials.
• Understanding how to manipulate these expressions with exponent rules is crucial for constructing and simplifying them.
• Algebraic expressions are the foundation of algebra and are essential in higher mathematics, requiring a firm grasp of each component and their operations.

These expressions are vital in forming equations and functions which model real-world problems.

Simplification in Algebra

Simplifying expressions in algebra is about making them as concise as possible while preserving their value. With our example \((-2)^{-1}\), simplification involves applying the negative exponent rule to rewrite the expression with positive exponents and then further refining it into its simplest form of \(\frac{1}{-2}\).

• One starts by identifying and applying necessary rules, like exponent or distributive rules, to transform the expression.
• Simplification can also imply combining like terms or factoring, depending on the expression's nature.
• The goal is to achieve an expression that is easier to understand and work with, particularly in solving equations or inequality.

Simplification of algebraic expressions is a fundamental skill that enhances problem-solving efficiency and accuracy across various areas of mathematics.