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Chapter 1: Problem 39

Find each quotient. $$ \frac{-32}{-4} $$

Short Answer

Expert verified
The quotient of \(\frac{-32}{-4}\) is 8.

Step by step solution

01

Identify the Dividend and Divisor

In the given quotient, \(\frac{-32}{-4}\), the dividend is -32 and the divisor is -4.
02

Divide the Absolute Values

First, ignore the negative signs and divide the absolute values of the numbers. Divide 32 by 4: \(\frac{32}{4} = 8\).
03

Determine the Sign

When both the dividend and the divisor have the same sign, the quotient is positive. Since both -32 and -4 are negative, their quotient is positive.
04

Combine the Results

From the previous steps, the absolute value of the quotient is 8, and the sign is positive. Thus, the quotient \(\frac{-32}{-4}\) is 8.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dividend and Divisor
In division, the dividend and divisor are the two main components. The dividend is the number being divided, while the divisor is the number you are dividing by. For example, in the expression \(\frac{-32}{-4}\), -32 is the dividend and -4 is the divisor. Identifying these correctly is crucial because it sets the stage for all subsequent steps in solving the division problem. Always ensure you know which is which to prevent mistakes in the division process.
Absolute Values
Absolute value represents the distance a number is from zero on the number line, regardless of direction. It is always a non-negative number. For example, the absolute value of both -32 and -4 is simply 32 and 4, respectively. The absolute value is denoted by vertical bars around the number, such as \(|-32| = 32\). To solve division problems, you first work with the absolute values because it simplifies the operation by removing any negative signs. Divide like usual: \(\frac{32}{4} = 8\).
Sign Rules in Division
The sign of the quotient in division depends on the signs of the dividend and the divisor:
  • If both numbers are positive or both are negative, the quotient is positive.
  • If one number is positive and the other is negative, the quotient is negative.
In our example, \(\frac{-32}{-4}\), both the dividend (-32) and the divisor (-4) are negative. Therefore, their quotient will be positive. Following the steps: first take the absolute values, perform the division, and then apply the proper sign based on these rules. Hence, the final quotient is positive 8.

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