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Order of Operations

Order of Operations

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The Importance of Following the Order of Operations in Simplifying Mathematical Expressions

When solving a mathematical expression, it is essential to follow a specific order of operations to ensure the correct answer. Failure to do so may result in an incorrect solution. Fortunately, there are tips and tricks to help remember the correct sequence of operations, including grouping symbols, exponents, multiplication, and division.

The Order of Operations

When faced with an equation containing multiple operations, there are four steps that can guide you in solving it correctly. First, evaluate any terms within grouping symbols, such as brackets or parentheses. Next, address any exponents or powers. Then, move on to multiplication and division, working from left to right. Lastly, complete any remaining addition and subtraction, also from left to right.

Types of Brackets

There are various types of brackets that may be used in mathematical expressions, including round brackets ( ), curly brackets { }, and box brackets [ ]. These symbols indicate which operations should be prioritized when solving an equation.

An Example of Solving an Equation using the Order of Operations

Let's walk through the application of the order of operations step by step with an example. Consider the following expression:

First, calculate the terms within the grouping symbols:

  • Calculate grouping symbols:

Next, address the exponent:

  • Calculate any exponents:

Then, move on to multiplication and division (if applicable):

  • Multiplication/division: This step is not needed in this expression.

Lastly, complete any addition and subtraction:

  • Addition/subtraction: Therefore, the answer is 42.

How to Remember the Correct Order of Operations

To aid in remembering the correct order of operations, the acronym PEMDAS can be utilized. It stands for Parentheses (grouping symbols), Exponents (powers and roots), Multiplication and Division, and Addition and Subtraction.

Examples of Order of Operations

Let's work through a few examples to further illustrate the order of operations.

Example 1:

Simplify

  • First, calculate the terms within the grouping symbols:
  • Calculate grouping symbols:
  • Next, address the exponent:
  • Calculate any exponents:
  • Then, move on to multiplication and division (if applicable):
  • Multiplication/division: This step is not needed in this expression.
  • Lastly, complete any addition and subtraction:
  • Addition/subtraction: Therefore, the answer is 42.

Example 2:

Simplify

  • First, calculate the terms within the grouping symbols:
  • Calculate grouping symbols:
  • Next, address the exponent:
  • Calculate any exponents:
  • Then, move on to multiplication and division (if applicable):
  • Multiplication/division: This step is not needed in this expression.
  • Lastly, complete any addition and subtraction:
  • Addition/subtraction: Therefore, the answer is 5.

Order of Operations with Algebra

The order of operations is equally important when working with algebraic expressions and solving for a given value of a variable.

An Example of Solving an Algebraic Expression Using the Order of Operations

Consider the expression . First, substitute 6 for the variable and then follow the order of operations using PEMDAS:

  • Next, address the exponent:
  • Calculate any exponents:
  • Then, move on to multiplication and division (if applicable):
  • Multiplication/division: This step is not needed in this expression.
  • Lastly, complete any addition and subtraction:
  • Addition/subtraction: Therefore when .

Similarly, when evaluating and , the correct order of operations should be followed:

  • Next, address the exponent:
  • Calculate any exponents:
  • Then, move on to multiplication and division (if applicable):
  • Multiplication/division: This step is not needed in these expressions.
  • Lastly, complete any addition and subtraction:
  • Addition/subtraction: Therefore, when and .
  • Lastly, complete any addition and subtraction:
  • Addition/subtraction: Therefore, when .

Key Takeaways

It is essential to follow a specific order of operations when evaluating both numerical and algebraic expressions to ensure the correct answer. The four steps to remember are: Grouped terms, parentheses, powers or exponents, multiplication or division (in order from left to right), and addition or subtraction (in order from left to right). PEMDAS is an acronym that can aid in remembering the correct order of operations. When dealing with algebra, it is also important to follow the order of operations when plugging in a given value for the variable in an equation.

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