Problem 16 - Algebra 1

Question

Evaluate the exponential expression. Write fractions in simplest form. $$\left(\frac{1}{5}\right)^{-1}$$

Step-by-Step Solution

Answer

The simplification of \((1/5)^{-1}\) is 5.

Step 1: Understand the Exponent Rule

The first step is to understand the rule about exponents. When a number is raised to the power of -1, it is as if taking the reciprocal of the number. That is, \(a^{-1} = 1/a\). In this case, \(a = 1/5 \) and the question is looking for \((1/5)^{-1}\).

Step 2: Apply the Rule

Next, apply the exponent rule to simplify the expression, yielding \(1/(1/5)\).

Step 3: Calculate the Reciprocal

From here, you can simplify the expression by dividing 1 by 1/5. Remember that dividing by a fraction is same as multiplying by its reciprocal. Hence, \(1/(1/5)\) can be rewritten as \(1*(5/1)\), which simplifies to 5.

Key Concepts

Negative Exponents

When dealing with negative exponents, it's important to understand what they imply for an expression. A negative exponent indicates that we are considering the reciprocal of the base raised to the positive equivalent of that exponent.

Here's a small breakdown:

• The term "exponent" refers to the power to which a number is raised.
• When the exponent is negative, such as in the expression \(a^{-n}\), it means we take the reciprocal: \(1/a^{n}\).
• This rule applies universally, making it a key concept for simplifying expressions with negative powers.

So, when you see an expression like \((1/5)^{-1}\), realize that it automatically tells you to find the reciprocal, which will be addressed more in depth in the following sections.

Reciprocal

The idea of a reciprocal is central when working with negative exponents. The reciprocal of a number is simply 1 divided by that number. It's like flipping the number around, especially when dealing with fractions. For example, the reciprocal of \(1/5\) is \(5/1\), which is simply 5.

Let's look at some key points:

• A reciprocal essentially inverts the fraction. In other words, the numerator and denominator switch places.
• This is particularly helpful when dividing by a fraction. Dividing by a fraction is equivalent to multiplying by its reciprocal.
• Thus, when you see \(1/(1/5)\), you know it turns into the reciprocal \(5/1\).

This understanding of reciprocals plays a vital role in simplifying expressions, particularly those involving negative exponents.

Simplifying Fractions

In mathematics, simplifying fractions involves reducing them to their simplest form, where the numerator and the denominator cannot be reduced any further while still being whole numbers. However, when working with expressions like \(1/(1/5)\), simplifying involves recognizing and manipulating reciprocal relationships.

Here’s how you simplify such fractions:

• Recognize that \(1/(1/5)\) is equivalent to multiplying by the reciprocal, transforming it into \(1 \times 5/1\).
• Since \(1 \times 5/1 = 5\), the expression simplifies neatly to a whole number, 5.
• These types of simplifications are straightforward once you understand the relationship between negative exponents and reciprocals.

Always ensure that your final result is in its simplest form for the clearest understanding of any mathematical problem.