Problem 43 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\frac{8}{s^{-1}}\)

Step-by-Step Solution

Answer

The expression simplifies to 8s.

Step 1: Understand the Problem

Our goal is to simplify the expression \( \frac{8}{s^{-1}} \). The fraction denominator contains the variable \( s \) raised to a negative exponent. Negative exponents will be addressed in the next step.

Step 2: Apply the Negative Exponent Rule

The rule for negative exponents states that \( a^{-b} = \frac{1}{a^{b}} \). Applying this rule, we can rewrite \( s^{-1} \) as \( \frac{1}{s} \).

Step 3: Simplify the Fraction

Replace \( s^{-1} \) in the denominator with \( \frac{1}{s} \). Thus, the expression becomes \( \frac{8}{\frac{1}{s}} \). Dividing by a fraction is equivalent to multiplying by its reciprocal.

Step 4: Multiply by the Reciprocal

Multiply 8 by the reciprocal of \( \frac{1}{s} \), which is \( s \). So, \( \frac{8}{\frac{1}{s}} \) simplifies to \( 8 \times s \).

Step 5: Write the Final Simplified Expression

The simplified expression is \( 8s \), as it does not have any negative exponents or fractions.

Key Concepts

Negative Exponents

A negative exponent is an expression where the exponent of a number is less than zero. It indicates the reciprocal, meaning the number is found in the denominator part when considered in a fraction form.\( a^{-b} \) means you have to take the reciprocal of \( a \) raised to the positive of \( b \), such that \( a^{-b} = \frac{1}{a^b} \). This rule helps convert a negative exponent into a positive one, making it easier to manipulate mathematically.
For example, \( s^{-1} \), as seen in the original exercise, transforms into \( \frac{1}{s} \). By understanding how to handle negative exponents, you can simplify expressions that might initially seem complicated.

Fraction Simplification

Fraction simplification is about making a fraction as simple as possible by reducing it, just like reducing a division problem into something more understandable. The goal is to have the simplest form without altering the value of the fraction. Simplifying fractions often involves canceling out common factors in the numerator and the denominator.
In the context of the original exercise, we started with \( \frac{8}{s^{-1}} \), and the negative exponent in the denominator was converted using the rule for negative exponents. As a result, \( \frac{8}{s^{-1}} \) becomes \( \frac{8}{\frac{1}{s}} \). To simplify it further, you need to recognize that dividing by a fraction is like multiplying by its reciprocal. This explains the critical step in switching from division to multiplication.

Reciprocal

The reciprocal of a number or a fraction is simply flipping the numerator and the denominator. If the original number is \( a \), its reciprocal will be \( \frac{1}{a} \). For a fraction like \( \frac{b}{a} \), the reciprocal is \( \frac{a}{b} \). The concept of reciprocals is crucial in division involving fractions, as dividing by a fraction means multiplying by its reciprocal.
Applying this to the exercise, once the negative exponent is converted, we need to use the reciprocal to simplify further. With \( \frac{8}{\frac{1}{s}} \), we multiply by the reciprocal of the fraction \( \frac{1}{s} \), which is \( s \). Thus, \( \frac{8}{\frac{1}{s}} \) simplifies to \( 8 \times s \). This action eliminates the fraction and negative exponent, leading to a simple expression of \( 8s \). Understanding reciprocals helps in making this transformation quick and straightforward.