Problem 8 - Elementary Algebra

Question

For Problems \(1-30\), evaluate each numerical expression. $$ \frac{1}{3^{-1}} $$

Step-by-Step Solution

Answer

The evaluated expression is 3.

Step 1: Understand the Negative Exponent

A negative exponent, such as in the expression \(3^{-1}\), indicates the reciprocal of the base raised to the positive exponent. Therefore, \(3^{-1}\) is the same as \(\frac{1}{3^1}\).

Step 2: Simplify the Reciprocal

Replace the expression \(3^{-1}\) with its reciprocal \(\frac{1}{3}\), so the original expression \(\frac{1}{3^{-1}}\) becomes \(\frac{1}{\frac{1}{3}}\). You now have a division of a fraction.

Step 3: Divide by the Fraction

Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, \(\frac{1}{\frac{1}{3}}\) can be rewritten as \(1 \times 3\), which simplifies to 3.

Key Concepts

Understanding Reciprocals

The term "reciprocal" might sound complex at first, but it refers to something quite simple in mathematics. The reciprocal of a number is essentially what you multiply that number by to get the result of 1. For any non-zero number, the reciprocal is 1 divided by that number.
Example: The reciprocal of 5 is \(\frac{1}{5}\). This is because when you multiply 5 by \(\frac{1}{5}\), you get 1.
Similarly, if you have a fraction, say \(\frac{2}{3}\), its reciprocal would be \(\frac{3}{2}\). All you need to do is flip the numerator and the denominator to find a fraction's reciprocal.

• Reciprocals turn numbers "inside out", making them very useful for division and handling negative exponents.
• Understanding reciprocals lets you simplify expressions and solve problems more easily.

In our exercise, we deal with the expression \(3^{-1}\), which means finding the reciprocal of 3. This is \(\frac{1}{3}\), showing how reciprocals connect with negative exponents.

Evaluating Numerical Expressions

Numerical expressions are mathematical phrases consisting of numbers and operations, like addition or multiplication. Evaluating a numerical expression means performing the operations in the correct order to reach a single final number.
In the context of negative exponents, these expressions often require a clear understanding of how to process them effectively.
Consider the expression \(\frac{1}{3^{-1}}\). Here's how you can approach it:

• Recognize what's asked: The expression involves a fraction and a negative exponent. This tells you that reciprocal handling will be necessary.
• Convert negative exponents: Switch the negative exponent to a positive by solving \(3^{-1}\) as \(\frac{1}{3}\). This small change simplifies the entire expression.

Once you manage these basic techniques, evaluating other expressions involving exponents becomes much smoother, leading to correct solutions and better understanding.

Simplifying Fraction Division

Dealing with fraction division might seem daunting at first, but it doesn't have to be. When you need to divide by a fraction, there's a simple rule: Multiply by the reciprocal.
This rule helps tackle expressions that involve dividing fractions, such as \(\frac{1}{\frac{1}{3}}\). Here's how to break it down:

• Identify the division: When the expression shows dividing by \(\frac{1}{3}\), note that you will multiply by its reciprocal instead.
• Find the reciprocal: Flip the fraction \(\frac{1}{3}\) to \(3\), as multiplying by \(3\) is equivalent to dividing by \(\frac{1}{3}\).
• Perform the multiplication: Now solve \(1 imes 3\). This simple multiplication results in 3.

The process of turning division into multiplication with the use of reciprocals streamlines solving such expressions. This approach not only simplifies calculations but also highlights the elegant symmetry within mathematics.