Problem 38 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(35-42,\) find the multiplicative inverse of each number. $$\frac{1}{7}$$

Step-by-Step Solution

Answer

The multiplicative inverse of \( \frac{1}{7} \) is \( 7 \).

Step 1: Identify the given number

The given number is the fraction \( \frac{1}{7} \).

Step 2: Find the multiplicative inverse of the given number

To find the multiplicative inverse of \( \frac{1}{7} \), reverse the numerator and denominator of \( \frac{1}{7} \) to get \( \frac{7}{1} \).

Step 3: Simplify the resulting fraction

Simplify \( \frac{7}{1} = 7 \). Therefore, the multiplicative inverse of \( \frac{1}{7} \) is \( 7 \).

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. They consist of two main components: the numerator and the denominator. The numerator is the top number, and it shows how many parts we have. The denominator is the bottom number, indicating the total number of equal parts the whole is divided into.
For example, in the fraction \(\frac{1}{7}\), "1" is the numerator, and "7" is the denominator. This fraction means we have 1 part of something divided into 7 equal parts.
Fractions can express quantities less than one (proper fractions), equal to one (such as \(\frac{7}{7}\)), or more than one (improper fractions like \(\frac{9}{7}\)). They're handy in both arithmetic and complex mathematical concepts like algebra.

Algebra and Its Basics

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It allows us to solve problems involving unknown values. By using variables, algebra helps us generalize mathematical operations and relationships.
An important aspect of algebra is understanding how different operations relate to each other. This includes knowing how to handle addition, subtraction, multiplication, and division with variables and constants.

• For example, in algebra, the expression \(x + 2 = 5\) helps us find the value of \(x\).
• Similarly, the concept of multiplicative inverses involves manipulating expressions to simplify equations or solve for unknowns.

Algebra provides the tools to understand various mathematical phenomena systematically, much like how understanding the multiplicative inverse helps in balancing equations.

Inverse Operations Explained

Inverse operations are operations that reverse the effect of each other. In the context of multiplication, finding an inverse means identifying a number which, when multiplied by the original number, yields 1.
To find the multiplicative inverse of a fraction, we simply flip the numerator and the denominator. For example, the multiplicative inverse of \(\frac{1}{7}\) is \(\frac{7}{1}\), or simply 7.
Finding inverses is crucial when solving equations. If you multiply a fraction by its inverse, you get 1, which is effectively like neutralizing the fraction in an equation.

• This concept is pivotal in algebra, particularly in equations where dividing by a fraction is needed.
• It's similar to how subtracting a number can be reversed by adding the same number, keeping equations balanced.

Understanding inverse operations makes many problems in mathematics easier to resolve, demonstrating the power of reversing operations strategically.