Problem 79 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Question: Rewrite the given expression using only positive exponents: $\frac{1}{a^{-1}}$ Solution: The rewritten expression with only positive exponents is $a$.

Step 1: Identify the negative exponent

First, let's identify the negative exponent in the expression. Here, we have a negative exponent (\(-1\)) in the denominator: $$ \frac{1}{a^{-1}} $$

Step 2: Use the exponent rule to make the exponent positive

To rewrite the expression using only positive exponents, we can use the exponent rule: \(a^{-n} = \frac{1}{a^n}\). Applying this rule to \(a^{-1}\), we get: $$ a^{-1} = \frac{1}{a^1} $$

Step 3: Substitute the new expression for the negative exponent

Now that we have a positive exponent for \(a^{-1}\), we can substitute this new expression back into the original expression: $$ \frac{1}{a^{-1}} = \frac{1}{\frac{1}{a^1}} $$

Step 4: Simplify the expression

To finalize the expression, we can simplify by multiplying both the numerator and denominator by \(a^1\) (since dividing by a fraction is the same as multiplying by its reciprocal): $$ \frac{1}{\frac{1}{a^1}} = \frac{1 \cdot a^1}{1} = a^1 $$

Step 5: Final Answer

So, the given expression with only positive exponents is: $$ a^1 \text{ or simply } a $$

Key Concepts

Negative Exponent Rule

Understanding the negative exponent rule is essential in algebra as it allows us to transform expressions with negative exponents into their equivalent forms with positive exponents. This rule indicates that any number with a negative exponent can be converted into its reciprocal raised to the opposite positive exponent. For example, given an expression with negative exponent like \(a^{-n}\), we can rewrite it as \(\frac{1}{a^n}\).

This transformation simplifies the process of working with exponential expressions, especially when dealing with complex equations or when simplifying algebraic expressions. It's crucial to remember that the base of the exponent doesn't change during this process; only the position relative to the fraction line (numerator or denominator) and the sign of the exponent are affected.

Exponential Expressions

Exponential expressions involve numbers or variables raised to a power, indicating repeated multiplication. They are represented as \(b^n\), where \(b\) is the base and \(n\) is the exponent. The exponent tells us how many times to multiply the base by itself. For example, \(3^4\) means 3 multiplied by itself 4 times (3 x 3 x 3 x 3).

When working with these expressions, especially in algebra, it's important to understand and apply the various properties of exponents, such as the product rule, quotient rule, power rule, and the negative exponent rule. Mastering these properties enables us to manipulate exponential expressions to simplify them or solve equations.

Algebraic Simplification

Algebraic simplification is the process of reducing an algebraic expression to its simplest form. This involves combining like terms, using the distributive property, and applying the rules of exponents to simplify expressions with powers. Algebraic simplification can make it easier to understand the core components of an expression and solve equations.

Simplifying an expression doesn't change its value; rather, it streamlines it for ease of use. For instance, the expression \(\frac{1}{a^{-1}}\) might appear complex at first glance, but using algebraic simplification, we see that it simplifies to just \(a\). It's a critical skill for students to develop as it applies to numerous areas in mathematics, from solving simple equations to tackling calculus problems.

Reciprocal of Exponents

The reciprocal of an exponential term involves inverting the base and switching the sign of the exponent. It plays a vital role when dealing with negative exponents. For example, the reciprocal of \(a^n\), where \(n\) is a positive integer, would be \(a^{-n}\) and vice versa.

When we encounter a term like \(a^{-n}\), we can think of it as the reciprocal of \(a^n\), which would be written as \(\frac{1}{a^n}\). This relationship between negative exponents and reciprocals is a key concept in algebra that aids in simplifying expressions and solving equations. It's important to grasp this concept to prevent errors when rewriting expressions and to achieve a better understanding of how exponents behave within mathematical operations.