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Chapter 5: Problem 151

Scientific calculators that have parentheses keys allow for the entry and computation of relatively complicated expressions in a single step. For example, the expression \(15+(10-7)^{2}\) can be evaluated by entering the following keystrokes: $$ 15+(10-7) y^{x} 2 = $$ Find the value of each expression in a single step on your scientific calculator. \((8-2) \cdot(3-9)\)

Short Answer

Expert verified
The value of the expression \((8-2) \cdot (3-9)\) is \(-36\).

Step by step solution

01

Simplify The Subtraction Inside Brackets

First, perform the subtraction operation inside the brackets. So, \(8 - 2 = 6\) and \(3 - 9 = -6\). The given expression now becomes \(6 \cdot -6\).
02

Carry Out Multiplication

Now, perform the multiplication of \(6\) (positive number) and \(-6\) (negative number). According to the multiplication rule, the multiplication of a positive number and a negative number gives a negative product, which is \(-36\)

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Key Concepts

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

Mathematical Expressions
Mathematical expressions consist of numbers, variables, operators, and sometimes brackets or parentheses. Each component has a specific function in the expression.
- **Numbers**: These are the actual quantities involved, such as 8, 2, or 3 in our example.
- **Operators**: Symbols like plus (+), minus (-), multiplication (\(\cdot\)), and division (\(/\)) indicate the operations to perform on the numbers.
- **Variables**: Although not present in every expression, these are letters representing numbers and allow generalization, primarily used in algebra.
Understanding the role of each component ensures accurate calculation and manipulation of expressions.
Parentheses in Calculations
Parentheses are essential when calculating complex mathematical expressions because they clarify the order of operations. They indicate which calculations should be grouped and performed first.
In our exercise, \((8-2) \cdot (3-9)\), the parentheses tell us to solve the subtraction operations before any multiplication. This priority ensures you get the correct result.When using scientific calculators, entering parentheses ensures that the grouped calculations are prioritized, which prevents errors during manual calculations.
Order of Operations
Order of operations is a set of rules to determine which procedures to perform first when evaluating a mathematical expression. An easy way to remember these rules is through the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In the given expression, \((8-2) \cdot (3-9)\), following the order of operations involves: - First, solving within the parentheses.- Then, performing multiplication between the results of the parentheses.Skipping any step or misunderstanding this order could lead to incorrect answers.
Multiplication Rules
The multiplication rules, especially when dealing with positive and negative numbers, are vital in mathematics. These rules dictate how numbers are combined:- A positive number multiplied by a positive number yields a positive product.- A positive number multiplied by a negative number results in a negative product.- A negative number multiplied by another negative gives a positive product.In our expression, \(6 \cdot -6\), the multiplication of a positive number by a negative number results in a negative product, which is \(-36\). Understanding and applying these rules correctly ensures accurate mathematical results.

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Most popular questions from this chapter

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-1000, r=0.1\)

Enough curiosities involving the Fibonacci sequence exist to warrant a flourishing Fibonacci Association. It publishes a quarterly journal. Do some research on the Fibonacci sequence by consulting the research department of your library or the Internet, and find one property that interests you. After doing this research, get together with your group to share these intriguing properties.

You will develop geometric sequences that model the population growth for California and Texas, the two most populated U.S. states. The table shows the population of California for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through \(2009 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} & \mathbf{2 0 0 5} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 33.87 & 34.21 & 34.55 & 34.90 & 35.25 & 35.60 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 36.00 & 36.36 & 36.72 & 37.09 & 37.25 \\ \hline \end{array} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that California has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling California's population, in millions, \(n\) years after \(1999 .\) c. Use your model from part (b) to project California's population, in millions, for the year 2020 . Round to two decimal places.

Write a formula for the general term (the nth term) of each arithmetic sequence. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. \(a_{1}=-20, d=-4\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(2,6,10,14, \ldots\)
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