Problem 56 - Beginning and Intermediate Algebra

Question

Find each reciprocal or multiplicative inverse. $$ -8 $$

Step-by-Step Solution

Answer

The reciprocal of -8 is \(-\frac{1}{8}\).

Step 1: Understanding Reciprocals

The reciprocal or multiplicative inverse of a number is a value that, when multiplied by the original number, results in 1. For any non-zero number 'a', its reciprocal is given as \( \frac{1}{a} \).

Step 2: Apply to -8

To find the reciprocal of \(-8\), we use the formula for reciprocals. Thus, the reciprocal of \(-8\) is \( \frac{1}{-8} \).

Step 3: Simplifying the Result

The expression \( \frac{1}{-8} \) is already simplified because there are no further simplifications needed for a single fraction with a numerator of 1. It is usually written and considered as is, i.e., \( -\frac{1}{8} \).

Key Concepts

Reciprocals

Reciprocals are a fundamental concept in algebra, with wide applications across various branches of mathematics. The reciprocal of a number is what you multiply that number by to get 1. For any number, say 'a', its reciprocal is represented as \( \frac{1}{a} \). This is only applicable for non-zero numbers, as dividing by zero is undefined.

Reciprocals work because when you multiply a number by its reciprocal, the result is always 1. For instance:

• If you have 5, its reciprocal is \( \frac{1}{5} \).
• Multiply them: \( 5 \times \frac{1}{5} = 1 \).

Finding reciprocals is essential in solving equations and inequalities, and it aids in the simplification of expressions.

Multiplicative Inverse

The multiplicative inverse is another term for reciprocal. This can sometimes cause confusion, but they mean exactly the same thing in algebra.

When we talk about the multiplicative inverse, we mean a number that, when multiplied with the original number, gives the identity element for multiplication, which is 1. For any real number except zero, the multiplicative inverse is found by taking the reciprocal. So, for a number like \(-8\), the multiplicative inverse is \( -\frac{1}{8} \).

Understanding the concept of a multiplicative inverse is crucial in algebra because it helps in finding solutions to equations and working with complex fractions. It always simplifies mathematical processes and fosters a deeper understanding of inverse operations.

Negative Numbers

Handling negative numbers is a vital part of algebra. Negative numbers are numbers less than zero, and they appear often in various mathematical contexts. When working with reciprocals or multiplicative inverses, dealing with negatives is important.

For example, the reciprocal of a negative number \(-8\) is \(-\frac{1}{8}\). Remember that dividing by a negative flips the sign of the result:

• If the original number is negative, so too will be its reciprocal.
• If you multiply \(-8\) by \(-\frac{1}{8}\), you'll notice the negatives cancel each other, leaving \(1\).

Understanding how to work with negative numbers ensures you can correctly apply reciprocals and inverses in various contexts, like solving equations and simplifying expressions correctly.