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Chapter 0: Problem 116

In Exercises \(111-120,\) use the order of operations to simplify each expression. $$8-3[-2(5-7)-5(4-2)]$$

Short Answer

Expert verified
The simplified form of the expression \(8-3[-2(5-7)-5(4-2)]\) is \(26\).

Step by step solution

01

Handle the inner brackets

Before performing any operations, simplify the terms inside the brackets. There are two subtractions inside the square brackets: \(5-7\) and \(4-2\). Simplify these to get: \(8-3[-2(-2)-5(2)]\)
02

Multiply negative numbers and constants

Next, perform the multiplication operations within the brackets. Multiply \(-2\) and \(-2\) to get \(4\), and multiply \(5\) and \(2\) to get \(10\), hence the expression is: \(8-3[4-10]\)
03

Subtract inside the square brackets

Simplify the expression inside the square brackets by subtracting \(10\) from \(4\) to get \(-6\), which transforms the expression to: \(8 - 3(-6)\)
04

Complete the last multiplication

Multiply \(3\) with \(-6\) to get \(-18\), hence the expression is: \(8 - (-18)\)
05

Simplify the subtraction operation

Finally, subtract \(-18\) from \(8\) to get \(26\).

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Key Concepts

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

Simplification
Simplification is the process of making a mathematical expression more manageable or easier to comprehend. It involves reducing an expression into its simplest form, applying the order of operations effectively. During simplification, we aim to eliminate complex terms and consolidate elements of the expression to reach a single value or a less complex expression.
This typically involves performing operations such as addition, subtraction, multiplication, and division in the correct order. Simplifying helps us understand the core parts of an expression, breaking down complicated problems into smaller, more digestible components.
Keep these key points in mind:
  • Always simplify terms inside the innermost brackets or parentheses first.
  • Follow the order of operations: Parentheses/Brackets, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • Look for like terms or expressions you can combine.
Brackets
Brackets are used in mathematics to specify the order in which operations should be performed. They indicate that the operations within them should be handled first, following the idea of priority in expressions.
In the expression given in the exercise, brackets help us determine which subtraction or multiplication to carry out first. The brackets ensure that we simplify the expression correctly by focusing on certain parts of the equation before others.
Understanding how to manage brackets involves:
  • Simplifying the terms within the innermost brackets first.
  • Ensuring even after the initial simplification, you observe which operations are still enclosed within brackets.
  • Carefully re-evaluating the expression after each simplification to see which parts of the problem remain inside brackets.
Multiplication
Multiplication is the operation of combining equal groups. In the context of this exercise, understanding how to handle multiplication accurately is key to simplifying the expression correctly.
Multiplication within expressions usually comes after managing brackets but before addition and subtraction. In our example, once the values inside the brackets are simplified, multiplication is performed to reduce these results down further.
Important multiplication reminders:
  • Remember the sign rules: Positive times positive or negative times negative results in a positive product, whereas positive times negative results in a negative product.
  • All multiplication should be completed inside the brackets prior to carrying out other operations like subtraction or addition.
  • It鈥檚 crucial to keep track of the signs during multiplication, as mixing these up can lead to incorrect results and confusion.
Subtraction
Subtraction in mathematics signifies the operation of removing one quantity from another. It plays a critical role when simplifying complex expressions as seen in this exercise.
Subtraction is often performed after multiplication in order of operations, simplifying expressions further. Additionally, in some cases, it involves working with negative numbers, which can sometimes behave counterintuitively.
When handling subtraction:
  • Ensure all other operations inside brackets are resolved prior to performing a subtraction outside of brackets.
  • Be cautious of subtracting negative numbers; reword this to adding a positive equivalent.
  • Verify your results by recalculating, especially when managing multiple layers of operations and combining different arithmetic operators.

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Most popular questions from this chapter

$$\text { Solve for } t: s=-16 t^{2}+v_{0} t$$

A company wants to increase the \(10 \%\) peroxide content of its product by adding pure peroxide (100\% peroxide). If \(x\) liters of pure peroxide are added to 500 liters of its \(10 \%\) solution, the concentration, \(C,\) of the new mixture is given by $$C=\frac{x+0.1(500)}{x+500}$$ How many liters of pure peroxide should be added to produce a new product that is \(28 \%\) peroxide?

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The bar graph shows the percentage of U.S. college freshmen with an average grade of A in high school. (GRAPH CAN NOT COPY) The data displayed by the bar graph can be described by the mathematical model $$p=\frac{4 x}{5}+25$$ where \(x\) is the number of years after 1980 and \(p\) is the percentage of U.S. college freshmen who had an average grade of A in high school. Use this information a. According to the formula, in 2010 , what percentage of U.S. college freshmen had an average grade of \(A\) in high school? Does this underestimate or overestimate the percent displayed by the bar graph? By how much? b. If trends shown by the formula continue, project when \(57 \%\) of U.S. college freshmen will have had an average grade of A in high school.

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