Problem 129 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. $$ \left(a^{-1}\right)^{-1} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression with only positive exponents: \(\left(a^{-1}\right)^{-1}\) Answer: \(a\)

Step 1: Identify the exponent equation.

We have the equation \(\left(a^{-1}\right)^{-1}\).

Step 2: Apply the Power Rule of Exponents

According to the power rule of exponents, when you raise a power to a power, you must multiply the exponents. In our case, we have \(a^{-1}\) raised to the power of \(-1\). So, we will now multiply the exponents: $$\left(a^{-1}\right)^{-1} = a^{-1 \times -1}$$

Step 3: Multiplying the Exponents

Let's multiply the exponents now: $$a^{-1 \times -1} = a^{1}$$

Step 4: Simplify the Expression

The simplified expression is \(a^1\). Since we don't need to show the exponent 1 when it's a positive value, we can just write it as \(a\). So, the final expression with only positive exponents is: $$\left(a^{-1}\right)^{-1} = a$$

Key Concepts

Understanding the Power Rule for Exponents

The power rule is a fundamental concept in mathematics, especially when dealing with exponents. It states that when you have an exponent raised to another exponent, you simply multiply them together. For instance, if you have

• \((x^m)^n\),

this becomes

• \(x^{m \times n}\).

This rule makes it easier to manage complex exponent expressions and helps simplify calculations sequentially.
In our exercise, we encountered \((a^{-1})^{-1}\). By applying the power rule, we transform the expression by multiplying the exponents:

• \(-1 \times -1 = 1\).

This operation eliminates complexity, enabling us to easily proceed to the solution

• \(a^1\).

Embracing the power rule greatly aids in mastering expressions with multiple layers of exponents.

Exploring Negative Exponents

Negative exponents can initially seem confusing, but they simplify math expressions in powerful ways. A negative exponent indicates reciprocal action. For instance,

• \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\).
• It turns a power into a fraction, moving the base from the numerator to the denominator.

By understanding this, you can tackle expressions confidently.
Our exercise began with an expression \((a^{-1})\). Here, \(a^{-1}\) indicates

• \(\frac{1}{a}\).

By implementing negative exponents, we open doors to more flexible rearrangements and simplifications of expressions. Remember, transforming negative exponents into their positive counterpart helps avoid intricate fractions and makes solving equations straightforward.

Simplifying Expressions Using Positive Exponents

Simplification of expressions to contain only positive exponents is not just about neatness; it aids in clearer communication of mathematical ideas. Simplified expressions are easier to understand and utilize in further calculations.
To simplify, focus on these steps:

• Identify unnecessary negative exponents.
• Apply rules such as the power rule to adjust the exponents to positive values.
• Condense the expression to its simplest form.

In our example, after applying the power rule, the initial expression \((a^{-1})^{-1}\) simplified to \(a^1\).
Ultimately, writing it as \(a\) eliminates any excess detail, making the solution not only concise but also ready for further manipulation or computation. Simplifying serves as a cornerstone for effective problem-solving and deeper understanding of algebraic concepts.