Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(35-42,\) find the multiplicative inverse of each number. $$-\frac{4}{9}$$

Step-by-Step Solution

Answer

The multiplicative inverse of \(-\frac{4}{9}\) is \(-\frac{9}{4}\).

Step 1: Identify the given number

The given number for which the multiplicative inverse is to be found is \(-\frac{4}{9}\).

Step 2: Find the multiplicative inverse (or reciprocal)

The multiplicative inverse of any number \(a\) is given by \(\frac{1}{a}\). Hence, the multiplicative inverse of \(-\frac{4}{9}\) will be \(-\frac{1}{(\frac{4}{9})}\). The aim is now to simplify the fraction within the fraction.

Step 3: Solve and Simplify

The simplification is achieved by swapping the numerator and the denominator, which basically means multiplying by the reciprocal of \(\frac{4}{9}\). So, \(-\frac{1}{\frac{4}{9}} = -\frac{1*9}{4} = -\frac{9}{4}\).

Key Concepts

Reciprocal

A reciprocal is a special type of number used in mathematics that, when multiplied by the original number, gives a result of one. To find the reciprocal of a fraction, simply swap its numerator and denominator. For example, the reciprocal of \(-\frac{4}{9}\) is \(-\frac{9}{4}\). Notice that the negative sign remains attached to the fraction. This operation is crucial when working with multiplicative inverses.
Reciprocals are helpful because they allow division to be transformed into multiplication. Instead of dividing by a fraction, you can multiply by its reciprocal. This can make mathematical operations much more straightforward. For example, dividing by \(\frac{4}{9}\) is the same as multiplying by \(\frac{9}{4}\).

• Swapping numerator and denominator gives you the reciprocal.
• Keep the negative sign if present.

Remember that for whole numbers, their reciprocal is \(\frac{1}{\text{number}}\). So, the reciprocal of 5 would be \(\frac{1}{5}\). It's an essential tool for simplifying mathematical problems.

Simplifying Fractions

Simplifying fractions makes them easier to work with by reducing them to their lowest terms. When you simplify a fraction, you're finding an equivalent fraction with the smallest possible numerator and denominator. Here's how you can simplify a fraction:
First, find the greatest common divisor (GCD) of the numerator and the denominator. Divide both the top and bottom numbers of the fraction by their GCD. This will reduce the fraction to its simplest form.

• Identify GCD of numerator and denominator.
• Divide both by the GCD.

For instance, if you have a fraction \(\frac{18}{24}\), you would find that the GCD is 6. After dividing both the numerator and the denominator by 6, the fraction simplifies to \(\frac{3}{4}\).
When dealing with reciprocals like \(-\frac{9}{4}\), check whether further simplification is possible. In this case, \(-\frac{9}{4}\) is already in its simplest form. Always ensure fractions are simplified for easier computation and comparison.

Inverse Operations

Inverse operations are mathematical operations that undo each other. For example, addition and subtraction are inverses because adding a number and then subtracting the same number brings you back to the original value. Multiplication and division have a similar relationship.
The concept of inverse operations is vital in solving equations and simplifying expressions. When you find the multiplicative inverse of a number, you're essentially finding the number that "undoes" the multiplication.

• Addition/Subtraction are inverse pairs.
• Multiplication/Division are inverse pairs.

In the context of this exercise, multiplying \(-\frac{4}{9}\) by its multiplicative inverse, \(-\frac{9}{4}\), results in 1. This concept helps in equation solving, as you can use inverse operations to isolate variables and simplify expressions.
Understanding these relationships allows you to approach problems more strategically and simplifies processes, especially when dealing with complex equations or algebraic expressions. Recognizing the inverse can be a powerful tool to unlock solutions efficiently.