Problem 110 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ 52 \div(-4)+(-32) \div(-8) $$

Step-by-Step Solution

Answer

-9

Step 1: Simplify the first division

Calculate the result of the first division in the expression. Simplify \( 52 \div (-4) \). This gives us \( 52 \div (-4) = -13 \).

Step 2: Simplify the second division

Next, simplify the second division. Calculate \( -32 \div (-8) \). This gives us \( -32 \div (-8) = 4 \).

Step 3: Combine the results

Finally, add the results from steps 1 and 2 together: \( -13 + 4 = -9 \).

Key Concepts

Integer Division

Understanding integer division is crucial when simplifying algebraic expressions. Integer division is the process of dividing one integer by another, and the result is also an integer. Depending on the numbers involved, the result can be positive, negative, or zero.

For instance, in the exercise, you simplified two divisions:

• The first was \( 52 \div (-4) = -13 \).
• The second was \( -32 \div (-8) = 4 \).

When dividing positive and negative numbers, follow these rules:

• 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.

These rules help ensure clarity and accuracy in operations involving integer division.

Negative Numbers

Negative numbers are numbers less than zero and are indicated by a minus sign (-). They are essential in algebra and many other areas of math.

In the exercise, you encountered negative numbers in both the divisor and the dividend. It's important to understand how to work with these numbers correctly:

• When you multiply or divide two numbers with the same sign, the result is positive.
• When you multiply or divide two numbers with different signs, the result is negative.

For example:

• In \( 52 \div (-4) \), you divided a positive by a negative, resulting in -13.
• In \( -32 \div (-8) \), you divided a negative by a negative, resulting in 4.

Correctly handling negative numbers will make simplifying expressions much simpler.

Step-by-step solution

A step-by-step approach helps to break down complex problems into more manageable parts. This method is particularly useful in algebra where multiple operations are often involved.

In the exercise, the solution was divided into three clear steps:

• First, you handled the division of \( 52 \div (-4) \), which yielded -13.
• Second, you performed \( -32 \div (-8) \), resulting in 4.
• Finally, you combined these results, adding -13 and 4 to get -9.

This step-by-step process ensures clarity and reduces the likelihood of errors. By focusing on one operation at a time and gradually building towards the final answer, students can better understand and solve complex algebraic expressions.