Problem 29 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Rewrite using the property of negative exponents

According to the property of negative exponents, we have: $$a^{-1} = \frac{1}{a^1}$$

Step 2: Simplify the expression

Since $$a^1$$ is just a, the expression becomes: $$\frac{1}{a}$$ So, the given expression $$a^{-1}$$ can be rewritten using only positive exponents as $$\frac{1}{a}$$.

Key Concepts

Exponential Expressions

Understanding exponential expressions is crucial for mastering various mathematical and real-world applications. An exponential expression is a mathematical phrase where a number, known as the base, is raised to a power, which is the exponent. The exponent signifies the number of times the base is multiplied by itself. For instance, in the expression \(2^3\), 2 is the base and 3 is the exponent, which tells us to multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\).

When it comes to the concepts of negative exponents, it's essential to recognize they follow a specific rule. An expression with a negative exponent, such as \(a^{-1}\), suggests division rather than multiplication. In essence, \(a^{-1}\) is equivalent to \(\frac{1}{a}\), where 'a' is any non-zero number. This concept allows us to transform any expression with negative exponents into a form with only positive exponents, facilitating easier computation and simplification.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra, which involves reducing an expression to its simplest form while retaining its original value. This often makes it easier to work with and understand. When it comes to simplifying exponential expressions, especially those with negative exponents, the process typically involves converting them to positive exponents using the rules of exponents.

Consider the expression \(a^{-1}\) as in the textbook exercise provided. The process of simplifying this expression starts with the realization that any non-zero number raised to the power of -1 is simply the reciprocal of that number. Therefore, \(a^{-1}\) becomes \(\frac{1}{a^1}\), which simplifies further to \(\frac{1}{a}\). This transformation facilitates understanding and using the expression in subsequent calculations, as working with positive exponents is generally more intuitive.

Properties of Exponents

The properties of exponents, also known as the laws of exponents, are a set of rules that govern how exponential terms are manipulated and combined. One of these key properties is the rule for negative exponents, which states that for any non-zero number 'a' and a positive integer 'n', the expression \(a^{-n}\) is equal to \(\frac{1}{a^n}\). In other words, a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent.

Other important properties include the product of powers rule, the power of a power rule, and the quotient of powers rule. For example, when multiplying expressions with the same base, you add the exponents \(a^m \times a^n = a^{m+n}\). Understanding these rules is essential for correctly simplifying and solving exponential expressions. They allow for a systematic approach to dealing with various exponential forms, ensuring accuracy and consistency in mathematical solutions.