Problem 62
Question
In Exercises \(29-72,\) use the order of operations to simplify each expression. $$4(-15)+|3(-10)|$$
Step-by-Step Solution
Verified Answer
The final answer is -30.
1Step 1: Parentheses
Solve the expressions inside the parentheses. For \(4(-15)\), multiply 4 and -15 to get -60. For \(3(-10)\), multiply 3 and -10 to get -30. The equation becomes \(-60 + |-30|\).
2Step 2: Absolute Value
Next, solve for the absolute value of -30, which is 30. The equation becomes \(-60 + 30\).
3Step 3: Addition
Finally, add -60 and 30 to get -30. The final answer is -30.
Key Concepts
Simplifying ExpressionsAbsolute ValueArithmetic Operations
Simplifying Expressions
Understanding the process of simplifying expressions is essential in mathematics, as it allows students to condense complex expressions into simpler forms. This is not only crucial for clearer communication but also for solving equations and evaluating mathematical statements.
To simplify an expression, one must follow a specific order, known as the order of operations. The order of operations, often remembered by the acronym PEMDAS, stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In the given exercise, the expression \(4(-15)+|3(-10)|\) is simplified by first addressing the operations inside the parentheses. Taking the example further, when you encounter multiple operations within an expression, reducing it down to the simplest form involves multiplication within the parentheses, followed by an understanding of absolute values, and concluded with addition or subtraction.
Incorporating exercise improvement advice, remember to always work through operations within parentheses first and check for possible simplifications before moving on to other operations. This step is critical and when missed, could lead to incorrect results.
To simplify an expression, one must follow a specific order, known as the order of operations. The order of operations, often remembered by the acronym PEMDAS, stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In the given exercise, the expression \(4(-15)+|3(-10)|\) is simplified by first addressing the operations inside the parentheses. Taking the example further, when you encounter multiple operations within an expression, reducing it down to the simplest form involves multiplication within the parentheses, followed by an understanding of absolute values, and concluded with addition or subtraction.
Incorporating exercise improvement advice, remember to always work through operations within parentheses first and check for possible simplifications before moving on to other operations. This step is critical and when missed, could lead to incorrect results.
Absolute Value
The absolute value of a number is a measure of its distance from zero on the number line, regardless of the direction. This means the absolute value of a number is always non-negative. For example, the absolute value of both \(-5\) and \(5\) is \(5\), because each is 5 units away from zero.
When simplifying expressions, it's important to correctly interpret the absolute value symbol. In our exercise, the absolute value of \(-30\) is \(30\), because we're looking at how far -30 is from zero, and not its sign.
A student’s mastery of absolute values will often determine the accuracy of their simplifications, especially when negative numbers are involved. Absolute values are key in understanding how to handle negative numbers in various operations, such as when simplifying the given expression \(-60 + |-30|\), where the absolute value changes the operation from a subtraction to an addition.
When simplifying expressions, it's important to correctly interpret the absolute value symbol. In our exercise, the absolute value of \(-30\) is \(30\), because we're looking at how far -30 is from zero, and not its sign.
A student’s mastery of absolute values will often determine the accuracy of their simplifications, especially when negative numbers are involved. Absolute values are key in understanding how to handle negative numbers in various operations, such as when simplifying the given expression \(-60 + |-30|\), where the absolute value changes the operation from a subtraction to an addition.
Arithmetic Operations
The term arithmetic operations typically refers to the most basic operations in mathematics, including addition, subtraction, multiplication, and division. These operations are the building blocks of all mathematical calculations and are performed following the order of operations to arrive at a correct solution.
In practicing these operations, always start with multiplication and division before moving on to addition and subtraction. For instance, in the provided exercise after simplifying the expressions in parentheses and applying the absolute value, the remaining arithmetic operation is addition: \(-60 + 30\), resulting in \(-30\).
When working on arithmetic problems, it is beneficial to be familiar with the properties of numbers, such as the distributive property, the associative property, and the commutative property, as they often provide shortcuts to simplifying expressions. By consistently practicing these principles, students can improve their skill in efficiently solving complex mathematical expressions.
In practicing these operations, always start with multiplication and division before moving on to addition and subtraction. For instance, in the provided exercise after simplifying the expressions in parentheses and applying the absolute value, the remaining arithmetic operation is addition: \(-60 + 30\), resulting in \(-30\).
When working on arithmetic problems, it is beneficial to be familiar with the properties of numbers, such as the distributive property, the associative property, and the commutative property, as they often provide shortcuts to simplifying expressions. By consistently practicing these principles, students can improve their skill in efficiently solving complex mathematical expressions.
Other exercises in this chapter
Problem 62
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