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Chapter 2: Problem 42

Simplify. See Examples 1 through \(10\). $$ |3-15| \cdot(-4) \div(-16) $$

Short Answer

Expert verified
The simplified result is 3.

Step by step solution

01

Simplify the Absolute Value

Calculate the absolute value of \( 3 - 15 \). Start by solving the expression inside the absolute value brackets. \( 3 - 15 = -12 \). The absolute value of \(-12\) is \(12\).
02

Substitute and Simplify the Expression

Replace \(|3-15|\) with \(12\) in the expression. Thus, the expression becomes \( 12 \cdot (-4) \div (-16) \).
03

Perform Multiplication

Multiply \(12\) by \(-4\) to get \(-48\). The expression is now \(-48 \div (-16)\).
04

Perform Division

Divide \(-48\) by \(-16\). Two negative signs will cancel out, so the result is a positive \(3\).

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Key Concepts

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

Absolute Value
When dealing with absolute values, we are essentially looking at the distance a number is from zero on a number line, without considering direction. This means that the absolute value of both \(+12\) and \(-12\) is \(|12| = 12\). It's important to first address any calculations inside the absolute value brackets before determining the absolute value.
  • Start by solving any expressions inside the absolute value signs.
  • Replace the negative result, if any, with its positive counterpart.
In our example, the expression \(3 - 15\) results in \'-12\'. Thus, \(|3-15| = 12\). Absolute value simplification always yields a non-negative number.
Multiplication
Multiplication is a basic arithmetic operation that combines numbers to get a product. In multiplication of integers, pay special attention to the signs involved:
  • Multiplying two positive numbers or two negative numbers results in a positive product.
  • Multiplying a positive number by a negative number, or vice versa, results in a negative product.
For the expression given, \(12 \cdot (-4)\), you multiply a positive number by a negative number. As per the rules, this results in a negative product, which is \'-48\'.
Division
Division can be thought of as distributing a number into a certain number of equal parts. Similar to multiplication, division of numbers follows rules regarding signs:
  • Dividing two numbers with the same sign yields a positive quotient.
  • Dividing two numbers with different signs yields a negative quotient.
In our exercise \(-48 \/ (-16)\), both numbers are negative. This means the negatives cancel out, resulting in a positive quotient. Therefore, the division results in \'3\'.
Negative Numbers
Negative numbers are numbers less than zero, usually represented with a minus sign. Handling negative numbers in operations involves understanding a few key points:
  • When adding two negative numbers, you get a more negative result.
  • When subtracting a negative number, it's like adding a positive number.
  • The product or quotient of two negative numbers is positive.
In the problem at hand, negatives played a crucial role especially in multiplication and division where they affected the sign of the result. Always keep an eye on how negatives might interact in calculations鈥攊t's a common source of mistakes!

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