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Chapter 1: Problem 8

In Exercises \(1-34,\) perform the indicated multiplication. $$(-11)(-1)$$

Short Answer

Expert verified
The answer is 11.

Step by step solution

01

Identify the numbers

The numbers presented are -11 and -1, which are both negative.
02

Apply the rule of multiplication for negative numbers

The rule is: A negative times a negative gives a positive. So (-11) times (-1) gives a positive number.
03

Multiply the numbers

Multiply the absolute values of the numbers, which are 11 and 1. The result is 11.

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Key Concepts

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

Negative Numbers
Negative numbers are numbers that are less than zero. They are typically represented with a minus sign before the number. For instance, in our example, \(-11\) and \(-1\) are negative numbers. This means they are both below zero on the number line.
Negative numbers are found in many real-world situations, like temperatures below freezing or debts in finance.
They are crucial in mathematics as they allow for more complex computations.
  • Negative numbers provide insight into directional changes. Going below zero on a number line or graphs may indicate downward or backward movement.
  • Operations with negative numbers often involve rules different from positive numbers. Understanding these rules is essential for solving equations and inequalities.
Multiplication Rule
In mathematics, the rule of multiplication for negative numbers is straightforward. When you multiply two negative numbers, the result is positive.
Here's how it works:
  • A negative number times a negative number always results in a positive number. For example, \((-11) \times (-1) = 11\).
  • If you multiply a positive number by a negative number, the result is negative. For example, \(5 \times (-3) = -15\).
The reason behind this rule is rooted in math's consistency with addition and subtraction patterns. If you think of multiplication as repeated addition, negative times negative can mean 'subtract the negative,' which results in a positive. This rule might seem abstract at first, but over time, with practice, it becomes intuitive.
Absolute Value
The absolute value of a number is its distance from zero on a number line, regardless of direction. Think of it as stripping away any negative sign to find a number's magnitude.
In the problem \((-11) \times (-1)\), we took the absolute values of \(-11\) and \(-1\). These are simply \(11\) and \(1\) respectively.
This is what we actually multiply during the calculation. Absolute value is always non-negative. It essentially tells us the size of a number without regard to its sign.
  • The absolute value of a negative number is positive. For example, \|-4| = 4\.
  • The absolute value of a positive number is the number itself. For example, \|7| = 7\.
Understanding absolute value is crucial, especially when dealing with real-life scenarios where magnitude matters and direction doesn't play a role.

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Most popular questions from this chapter

In Palo Alto, California, a government agency ordered computer-related companies to contribute to a pool of money to clean up underground water supplies. (The companies had stored toxic chemicals in leaking underground containers.) The mathematical model $$ C=\frac{200 x}{100-x} $$ describes the cost, \(C,\) in tens of thousands of dollars, for removing \(x\) percent of the contaminants. Use this formula to solve. a. Find the cost, in tens of thousands of dollars, for removing \(60 \%\) of the contaminants. b. Find the cost, in tens of thousands of dollars, for removing \(90 \%\) of the contaminants. c. Describe what is happening to the cost of the cleanup as the percentage of contaminants removed increases.

In Exercises \(147-149,\) perform the indicated operation. $$(-6)^{2}=(-6)(-6)=?$$

Exercises \(150-152\) will help you prepare for the material covered in the next section. In each exercise, an expression with an exponent is written as a repeated multiplication. Find this product, indicated by a question mark. $$(-2)^{4}=(-2)(-2)(-2)(-2)=?$$

In Palo Alto, California, a government agency ordered computer-related companies to contribute to a pool of money to clean up underground water supplies. (The companies had stored toxic chemicals in leaking underground containers.) The mathematical model $$ C=\frac{200 x}{100-x} $$ describes the cost, \(C,\) in tens of thousands of dollars, for removing \(x\) percent of the contaminants. Use this formula to solve. a. Find the cost, in tens of thousands of dollars, for removing \(50 \%\) of the contaminants. b. Find the cost, in tens of thousands of dollars, for removing \(80 \%\) of the contaminants. c. Describe what is happening to the cost of the cleanup as the percentage of contaminant removed increases.

Use a calculator to find a decimal approximation for each irrational number, correct to three decimal places. Between which two integers should you graph each of these numbers on the number line? $$2-\sqrt{5}$$
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