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Chapter 1: Problem 24

Simplify each numerical expression. (Objective 2) $$14 \div 7 \cdot 8-35 \div 7 \cdot 2$$

Short Answer

Expert verified
The simplified expression is 6.

Step by step solution

01

Perform Division Operations

First, divide 14 by 7, and 35 by 7 separately. \( 14 \div 7 = 2 \) and \( 35 \div 7 = 5 \). After completing the divisions, the expression becomes: \( 2 \cdot 8 - 5 \cdot 2 \).
02

Perform Multiplication Operations

Next, multiply the results from Step 1 by their respective numbers: \( 2 \cdot 8 = 16 \) and \( 5 \cdot 2 = 10 \). The expression now becomes: \( 16 - 10 \).
03

Perform Subtraction

Finally, subtract 10 from 16: \( 16 - 10 = 6 \).

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Key Concepts

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

Order of Operations
When simplifying numerical expressions, it's crucial to perform operations in the correct order. This is often remembered using the acronym PEMDAS:
  • P: Parentheses
  • E: Exponents
  • M and D: Multiplication and Division (from left to right)
  • A and S: Addition and Subtraction (from left to right)
In the expression \(14 \div 7 \cdot 8 - 35 \div 7 \cdot 2\), we focus on the division and multiplication first.
Follow the rule of performing these operations from left to right as they appear. This ensures that you maintain the integrity of the expression's value.
Division
Division is one of the four basic arithmetic operations, and in our expression, we start by dividing. The operation "\(\div\)" is straightforward.
  • When dividing, you are essentially distributing a total into equal parts.
  • The division in our expression involves both \(14 \div 7\) and \(35 \div 7\).
Each of these divisions simplifies as follows:
\(14 \div 7 = 2\) since 14 is split into 7 equal pieces, each worth 2.
Similarly, \(35 \div 7 = 5\) as 35 is split into 7 equal pieces, each 5 in value.
Simplifying division early helps make other operations like multiplication easier to handle.
Multiplication
After division comes multiplication, acting upon the results derived earlier in the simplification process.
  • This operation involves combining groups of a certain number together.
  • In our problem, multiply \(2 \cdot 8\) and \(5 \cdot 2\).
  • \(2 \cdot 8 = 16\): If you have 2 sets of 8, they collectively form 16.
  • \(5 \cdot 2 = 10\): Here, 5 sets of 2 give a total value of 10.
Multiplication carries significant weight in transforming the expression and is performed left to right, after addressing any division.
Subtraction
Finally, we reach the subtraction step, which simplifies the expression to its final form.
  • Subtraction finds the difference between two numbers.
  • In our expression, it's represented by the "\(-\)" symbol.
Here, subtract 10 from 16, which results in 6:
\(16 - 10 = 6\).
Subtraction is typically the last step in simplification, where the previous results combine to form the final answer.
Understanding how to navigate subtraction correctly ensures accuracy in the final result of any simplified numerical expression.

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Most popular questions from this chapter

Simplify each algebraic expression by combining similar terms. $$12 x-14 x+x$$

Simplify each numerical expression. Don't forget to take advantage of the properties if they can be used to simplify the computation. $$36-48-22+41$$

Simplify each algebraic expression by combining similar terms. $$-3(x+2)-4(x-10)$$

Find the value of \(\frac{h\left(b_{1}+b_{2}\right)}{2}\) for each set of values for the variables \(h, b_{1}\), and \(b_{2}\). (Subscripts are used to indicate that \(b_{1}\) and \(b_{2}\) are different variables.) \(h=8, b_{1}=17\), and \(b_{2}=24\)

State in your own words the associative property for addition of integers.
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