Problem 39 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ \frac{-45}{-5} $$

Step-by-Step Solution

Answer

Answer: The value of the expression is 9.

Step 1: Identify the given expression

We have the expression: $$ \frac{-45}{-5} $$

Step 2: Apply the rule for dividing integers

When dividing two negative integers, the result is positive. So, we can rewrite the expression as: $$ \frac{45}{5} $$

Step 3: Divide the numerator by the denominator

Now, we simply divide the numerator by the denominator: $$ \frac{45}{5} = 9 $$ The value of the given expression is 9.

Key Concepts

Negative Numbers

Negative numbers are values that are less than zero and are found on the left side of the number line. They represent quantities smaller than zero, often indicating a lack of something or a direction opposite to the positive. For instance, a negative temperature might indicate freezing cold.

When working with negative numbers in arithmetic operations, such as addition, subtraction, multiplication, and division, certain rules apply.

• Adding a negative number is the same as subtracting its positive counterpart.
• Multiplying or dividing two negative numbers results in a positive number.
• If one number is negative and one is positive in multiplication or division, the result is always negative.

Understanding these rules is crucial for correctly solving problems involving negative numbers.

Expression Simplification

Expression simplification involves reducing an arithmetic expression into its simplest form. This makes calculations easier and comparisons more straightforward. The process may involve removing parentheses, combining like terms, or performing operations.

Simplifying an expression like \(\frac{-45}{-5}\) involves recognizing that division of two negative numbers yields a positive result. This reduces the expression to \(\frac{45}{5}\).

Key steps in expression simplification often include:

• Identifying and removing unnecessary elements, such as extra zeros or identical terms on either side of an equation.
• Reordering terms to make calculations clearer and simpler.
• Applying arithmetic operations to combine or eliminate terms.

These steps can greatly reduce the complexity of mathematical problems.

Numerator and Denominator

The terms numerator and denominator are parts of a fraction. The numerator is the top number and represents how many parts of a whole are being considered. The denominator is the bottom number and indicates into how many parts the whole is divided.

In the fraction \(\frac{-45}{-5}\), \(-45\) is the numerator and \(-5\) is the denominator. Each plays an essential role in determining the value of the fraction.

Simplifying a fraction involves dividing the numerator by the denominator, which can result in either a whole number or a simplified fraction. This is crucial in understanding the relationship between the two and solving fractional expressions correctly.

• Always perform the division as the final step for accurate results.
• Consider the roles of negative or positive signs carefully.

Integer Arithmetic

Integer arithmetic deals with operations involving whole numbers, which include positive numbers, negative numbers, and zero. Numbers in integer arithmetic have no fractional or decimal components.

A fundamental aspect of integer arithmetic is understanding how basic operations affect these numbers, particularly how division is treated, as seen in the problem \(\frac{-45}{-5}\). The negative signs cancel each other out, resulting in a positive quotient.

• Adding and subtracting integers follow the rules of adding and cancelling out like and unlike signs.
• Multiplication and division have rules for determining the sign of the result (same signs yield positive, different signs yield negative results).

Mastery of integer arithmetic ensures accuracy in calculations across various mathematical problems.