Problem 41 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(35-42,\) find the multiplicative inverse of each number. $$-\frac{2}{5}$$

Step-by-Step Solution

Answer

The multiplicative inverse of \(-\frac{2}{5}\) is \(-\frac{5}{2}\).

Step 1: Understand the concept of the multiplicative inverse.

Knowing that the multiplicative inverse of a number is 1 divided by that number, the multiplicative inverse of \(-\frac{2}{5}\) can be written as \(1/-\frac{2}{5}\).

Step 2: Solve the division operation.

In order to divide by a fraction, we can multiply by its reciprocal (flip the fraction). So this becomes \(-1 * \frac{5}{2}\).

Step 3: Perform the multiplication

The multiplication of the two numbers gives \(-\frac{5}{2}\), which is the multiplicative inverse of \(-\frac{2}{5}\).

Key Concepts

Understanding Algebraic Operations

Algebraic operations are the foundation of algebra and include addition, subtraction, multiplication, and division, as well as the operations involving exponents and roots. When dealing with algebraic expressions, it's crucial to understand the rules and properties that govern these operations, such as the commutative, associative, and distributive properties. These operations help to solve equations and manipulate expressions to simplify them or rearrange them into a more usable form.

In the context of finding the multiplicative inverse, we utilize multiplication and division operations predominantly. For example, in our exercise, to find the multiplicative inverse of \( -\frac{2}{5} \) we performed division which was then simplified using multiplication. It's essential to keep these operations straight, as confusing them can lead to incorrect solutions. Always remember, the multiplicative inverse is what you multiply a number by to get the identity element for multiplication, which is 1.

The Reciprocal of a Number

The reciprocal of a number, also known as its multiplicative inverse, is one of the key concepts in fraction division and algebra. To find the reciprocal, we simply swap the numerator and the denominator of the fraction. In the case of an integer or a whole number 'a', the reciprocal is \( \frac{1}{a} \). It's important to note that the reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \), provided that neither 'a' nor 'b' is zero, since division by zero is undefined.

For our sample exercise, the reciprocal of \( -\frac{2}{5} \) is \( -\frac{5}{2} \). Understanding the concept of a reciprocal is crucial as it is used in operations involving fraction division and solving equations involving fractions.

Performing Fraction Division

Fraction division can be a tricky concept to master, but it is made much simpler by using the reciprocal. When we divide by a fraction, we are effectively multiplying by its reciprocal. This rule complies with the idea that dividing by a number is the same as multiplying by its multiplicative inverse.

To solve a fraction division problem like \( 1 / -\frac{2}{5} \), we actually convert it to \( 1 \cdot -\frac{5}{2} \). The reason for this conversion is that multiplication is often easier to carry out than division, especially when fractions are involved. When you're asked to divide by a fraction, always remember to multiply by its reciprocal instead. This will simplify the operation and help you reach the solution more efficiently, as shown in our example where the multiplicative inverse of \( -\frac{2}{5} \) was found to be \( -\frac{5}{2} \).