Problem 36 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(35-42,\) find the multiplicative inverse of each number. $$3$$

Step-by-Step Solution

Answer

The multiplicative inverse of 3 is \(1/3\).

Step 1: Understanding the multiplicative inverse

The multiplicative inverse of a number is simply another number, which when multiplied with the original number, gives the product as 1. So, to find out the multiplicative inverse of a number \(n\), we simply calculate \(1/n\).

Step 2: Apply multiplicative inverse formula

Applying this formula to the number 3, it will be \(1/3\). So, the multiplicative inverse of 3 is \(1/3\).

Key Concepts

Multiplicative Inverse Formula

Understanding the concept of a multiplicative inverse is crucial in many areas of mathematics, particularly when dealing with division and rational numbers. The multiplicative inverse of a number, commonly referred to as its reciprocal, is a value that, when multiplied by the original number, results in the identity element of multiplication, which is 1.

For any non-zero number 'a', the multiplicative inverse is given by the formula \( a^{-1} = \frac{1}{a} \). For example, the inverse of 3, as per this formula, is simply computed as \( 3^{-1} = \frac{1}{3} \), which means that when 3 is multiplied by its multiplicative inverse \(\frac{1}{3}\), the result is 1. It is important to note that the multiplicative inverse of zero does not exist since division by zero is undefined. This concept is a cornerstone in algebra and is widely used to solve equations and in various applications involving ratios and proportions.

Algebraic Operations

Algebraic operations are the building blocks for manipulating and solving algebraic expressions and equations. They consist of the basic arithmetic operations such as addition, subtraction, multiplication, and division, extended to variables, exponents, and more complex structures.

In the context of our problem, multiplication is the key operation since it is directly involved with the concept of multiplicative inverses. Algebra emphasizes the use of properties of operations, like the commutative property of multiplication which states that the order in which two numbers are multiplied does not affect the product, and the multiplicative identity property where any number multiplied by 1 equals itself. By mastering these operations and their properties, students are able to manipulate algebraic expressions and solve equations effectively.

Inverse Operations

Inverse operations are operations that undo each other. They are fundamental in solving algebraic equations and performing arithmetic computations. The most common pairs of inverse operations are addition and subtraction, as well as multiplication and division.

When it comes to multiplicative inverses, the act of multiplying by a number's inverse is the same as dividing by the number itself. So, the inverse operation to multiplication in this case is division. This becomes especially useful when we want to solve for an unknown variable in an equation. If the variable is multiplied by a certain number, we can perform the inverse operation (division by that number's inverse) to isolate and solve for the variable. Understanding inverse operations is essential for balancing equations, simplifying expressions, and finding solutions to mathematical problems.