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Chapter 1: Problem 75

Simplify each numerical expression. $$56 \div(-8)-(-6) \div(-2)$$

Short Answer

Expert verified
The simplified expression is -10.

Step by step solution

01

Simplifying the first division

Start with the first division in the expression: \( 56 \div (-8) \). We divide 56 by -8, which leads to a result of -7 because dividing a positive number by a negative number gives a negative result. Thus, \( 56 \div (-8) = -7 \).
02

Simplifying the second division

Now, move to the second division: \((-6) \div (-2)\). Here, dividing a negative number by another negative number results in a positive number. Therefore, \((-6) \div (-2) = 3\).
03

Combining results

Combine the results from the previous steps: \(-7 - 3\). This simplifies the expression to \(-7 - 3 = -10\).

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Key Concepts

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

Division of Integers
When you divide integers, you need to consider the signs of the numbers involved. An integer is a whole number that can be positive, negative, or zero. While zero cannot act as a divisor, it can be a dividend.
  • Positive divided by Negative: When you divide a positive integer by a negative integer, the result is always negative. For example, using the division in the expression, \( 56 \div (-8) = -7 \).
  • Negative divided by Negative: Dividing a negative integer by another negative integer results in a positive integer. This is illustrated by \( (-6) \div (-2) = 3 \).
  • Division by Zero: Remember, division by zero is undefined. So, never use zero as a divisor.
Recognizing these patterns helps in simplifying numerical expressions involving division.
Simplification Steps
Simplifying numerical expressions involves working through the operations one step at a time. Prioritize operations involving brackets and exponents first, but in this case, we focus on divisions.
Follow these steps:
  • Identify Divisions: Locate all divisions in the expression. This example has two divisions: \( 56 \div (-8) \) and \((-6) \div (-2) \).
  • Perform Each Division: Conduct each division separately. First, \( 56 \div (-8) = -7 \) and then, \((-6) \div (-2) = 3 \).
  • Combine Results: After calculating the results of the divisions, simplify by combining them. Here, it鈥檚 \(-7 - 3 = -10 \).
Breaking down the expression into manageable parts ensures easier processing and accurate results.
Negative Numbers in Arithmetic
Negative numbers can sometimes confuse learners when used in arithmetic operations. It's essential to grasp how they affect addition, subtraction, multiplication, and division.
Here are some guidelines:
  • Adding and Subtracting Negatives: Adding a negative is similar to subtracting a positive. Conversely, subtracting a negative could be seen as adding a positive. For example, \(-7 - 3 = -10\) uses this logic.
  • Multiplying and Dividing: With multiplication and division, the product or quotient of two numbers with the same signs is positive, while different signs yield a negative.
  • Practical Insight: Understanding negative numbers helps in real-world contexts, like finances where negative values might represent debts or losses.
Gaining comfort with negative numbers empowers you to tackle arithmetic challenges confidently.

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Most popular questions from this chapter

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$5(2 x+1)+4(3 x-2)$$

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. Six more than one-third of a number

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(3 a^{2}+2 b^{2}, \quad a=2\) and \(b=5\)

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$4\left(n^{2}+3\right)+\left(n^{2}-7\right)$$

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$-(3 x-1)-2(5 x-1)+4(-2 x-3)$$
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