Problem 18 - Intermediate Algebra

Question

Perform the following operations with real numbers. $$ (-81) \div(-3) $$

Step-by-Step Solution

Answer

The result of \((-81) \div (-3)\) is 27.

Step 1: Understand the Operation

The expression \((-81) \div (-3)\) tells us to divide \(-81\) by \(-3\). We need to figure out how many times \(-3\) goes into \(-81\).

Step 2: Examine the Signs

Consider the signs of the numbers involved. Here, both numbers are negative. When dividing two numbers with the same sign, the result is positive.

Step 3: Calculate the Absolute Values

Find the absolute values of \(-81\) and \(-3\). These are just \(81\) and \(3\), respectively.

Step 4: Perform Division with Absolute Values

Divide the absolute values: \(81 \div 3 = 27\).

Step 5: Apply the Sign Rule

Since both the dividend and the divisor were negative, according to Step 2, the result should be positive. So, the final answer is \(27\).

Key Concepts

Division

Division is one of the fundamental operations in mathematics, where a number (the dividend) is split into equal parts by another number (the divisor). In the operation \((-81) \div (-3)\), we start by identifying our dividend, which is \(-81\), and our divisor, which is \(-3\). The result of this division will tell us how many times \(-3\) fits into \(-81\).

• **Dividend**: The number you want to divide. Here, it is \(-81\).
• **Divisor**: The number by which you divide. In this example, it is \(-3\).
• **Quotient**: The result of the division. Our goal is to find this value.

When performing division, it is essential to evaluate both the absolute values of the numbers involved and their signs to ensure the result is accurate.

Absolute Value

The absolute value of a number is essentially its magnitude without considering its sign. This means for any negative number, its absolute value is the same number but positive.
In the example \((-81) \div (-3)\), we find the absolute values before dividing:

• The absolute value of \(-81\) is \(81\).
• The absolute value of \(-3\) is \(3\).

Understanding absolute value is crucial because it allows us to work with the magnitude of numbers independently of their sign, providing a straightforward approach to division.

Sign Rules

When we divide or multiply real numbers, the sign rules are essential to determine whether the result is positive or negative.
In any given operation:

• If both numbers have the same sign (both positive or both negative), the result is positive.
• If the numbers have different signs (one positive and one negative), the result is negative.

For our example \((-81) \div (-3)\), both \(-81\) and \(-3\) have the same sign (both negative). Therefore, according to the sign rules, the quotient will be positive.
This rule helps simplify the process of finding the sign of the result, especially when working with large sets of numbers, by merely observing their initial signs.