Question

Prep Exercise 6 When dividing two num bers that have the same sign, the result is ___. Prep Exercise 7 When dividing two numbers that have different signs, the result is ___. divide. $$-48 \div 8$$

Short answer

Answer: The result of $$-48 \div 8$$ is $$-6$$.

Step-by-step solution

Prep Exercise 6: Division with same signs

When dividing two numbers with the same sign (both positive or both negative), the result is positive.

Prep Exercise 7: Division with different signs

When dividing two numbers with different signs (one positive and one negative), the result is negative.

Identify the signs of the numbers in the given division problem

We have to divide $$-48 \div 8$$. Here, -48 is a negative number, and 8 is a positive number. Hence, they have different signs.

Apply the rule for division with different signs

Since the numbers have different signs, using the outcome from Prep Exercise 7, we know that the result of the division will be negative.

Perform the division

Now, we will divide 48 by 8 (ignoring the negative sign for now): $$\frac{48}{8}= 6$$.

Apply the sign to the result

Since we already know the result is negative, the final result of the division is: $$-48 \div 8 = -6$$.

Key concepts

Positive and Negative Numbers

Understanding positive and negative numbers is essential in arithmetic. In essence, positive numbers are those without a negative sign, typically representing counts or measurements like distances, lengths, and quantities that can only be zero or above. Negative numbers, which have a minus sign (-), represent the opposite, such as debts or temperatures below freezing.

When working with positive and negative numbers, imagine a number line: positive numbers sit to the right of zero, while negative numbers lie to the left. Moving left from zero leads you into the realm of negative numbers, whereas moving right keeps you with the positives. When we combine these numbers through operations like division, as in our exercise, the 'rules of signs' come into play to determine the result's sign.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. These are the building blocks of mathematics and are used to perform calculations.

Each operation has its own set of rules for handling numbers, particularly when it comes to combining positive and negative numbers. For instance, with division, we are essentially determining how many times one number is contained within another. In our exercise, the division of -48 by 8 was examined. To find a solution, first, we ignored the signs to divide the absolute values, and then we applied the 'rules of signs' to determine the sign of the result.

Rules of Signs

The 'rules of signs' are fundamental when performing arithmetic operations with integers. They help us to predict the sign of the result:

• If two positive numbers are divided, the result is positive.
• If two negative numbers are divided, the result is also positive because the negatives cancel each other out.
• However, when a positive and a negative number are divided, or vice versa, the result is negative.

Applying this to our exercise, -48 (negative) divided by 8 (positive) results in a negative number. Once we've divided the absolute values (48 by 8, which is 6), we attach the negative sign as dictated by the 'rules of signs', giving us -6 as the final answer.