Identify The Grouping Symbols That Represent The Second Step In The Evaluation Of The Numerical Expression:$\[ 2\{1+[4(2+1)]+3\} \\]

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Introduction

In mathematics, grouping symbols are used to clarify the order of operations in numerical expressions. These symbols, such as parentheses, brackets, and braces, help to ensure that mathematical expressions are evaluated correctly. In this article, we will focus on identifying the grouping symbols that represent the second step in the evaluation of the numerical expression: ${ 2{1+[4(2+1)]+3} }..

The Order of Operations

The order of operations is a set of rules that dictate the order in which mathematical operations should be performed in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Grouping Symbols

Grouping symbols are used to group parts of an expression together, making it easier to evaluate the expression. The most common grouping symbols are:

  • Parentheses ( ): Used to group expressions that need to be evaluated first.
  • Brackets [ ]: Used to group expressions that need to be evaluated next.
  • Braces { }: Used to group expressions that need to be evaluated last.

Evaluating the Numerical Expression

Now that we have a basic understanding of grouping symbols and the order of operations, let's evaluate the numerical expression: ${ 2{1+[4(2+1)]+3} }..

Step 1: Evaluate the Expression Inside the Parentheses

The first step is to evaluate the expression inside the parentheses: 2+12+1. This expression is evaluated first because it is inside the parentheses.

2+1=3{ 2+1 = 3 }

Step 2: Evaluate the Expression Inside the Brackets

The next step is to evaluate the expression inside the brackets: 4(3)+34(3)+3. This expression is evaluated next because it is inside the brackets.

4(3)=12{ 4(3) = 12 } 12+3=15{ 12 + 3 = 15 }

Step 3: Evaluate the Expression Inside the Braces

The final step is to evaluate the expression inside the braces: 1+15+31+15+3. This expression is evaluated last because it is inside the braces.

1+15=16{ 1 + 15 = 16 } 16+3=19{ 16 + 3 = 19 }

Step 4: Multiply the Result by 2

The final step is to multiply the result by 2.

2×19=38{ 2 \times 19 = 38 }

Conclusion

In conclusion, the grouping symbols that represent the second step in the evaluation of the numerical expression ${ 2{1+[4(2+1)]+3} }$ are the brackets [ ]. The brackets are used to group the expression 4(2+1)+34(2+1)+3 together, making it easier to evaluate the expression. By following the order of operations and using the correct grouping symbols, we can evaluate the numerical expression correctly.

Common Mistakes

When evaluating numerical expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not following the order of operations: Make sure to evaluate expressions inside parentheses first, then exponents, then multiplication and division, and finally addition and subtraction.
  • Not using the correct grouping symbols: Make sure to use the correct grouping symbols, such as parentheses, brackets, and braces, to group expressions together.
  • Not evaluating expressions inside grouping symbols: Make sure to evaluate expressions inside grouping symbols before moving on to the next step.

Practice Problems

Here are some practice problems to help you understand grouping symbols and the order of operations:

  1. Evaluate the numerical expression: ${ 3{2+[5(4-2)]+1} }..
  2. Evaluate the numerical expression: ${ 4{1+[2(3+1)]+2} }..
  3. Evaluate the numerical expression: ${ 2{3+[4(2-1)]+1} }..

Answer Key

  1. { 3\{2+[5(2)]+1\} }$ = 3\{2+10+1\} = 3\{13\} = 39 \]

  2. { 4\{1+[2(4)]+2\} }$ = 4\{1+8+2\} = 4\{11\} = 44 \]

  3. { 2\{3+[4(1)]+1\} }$ = 2\{3+4+1\} = 2\{8\} = 16 \]

Frequently Asked Questions

Q: What are grouping symbols? A: Grouping symbols are used to group parts of a mathematical expression together, making it easier to evaluate the expression. The most common grouping symbols are parentheses ( ), brackets [ ], and braces { }.

Q: Why are grouping symbols important? A: Grouping symbols are important because they help to clarify the order of operations in a mathematical expression. By using the correct grouping symbols, you can ensure that mathematical expressions are evaluated correctly.

Q: What is the order of operations? A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I use parentheses in a mathematical expression? A: To use parentheses in a mathematical expression, simply place the expression you want to evaluate first inside the parentheses. For example: ${ 2(3+1) }$.

Q: How do I use brackets in a mathematical expression? A: To use brackets in a mathematical expression, simply place the expression you want to evaluate next inside the brackets. For example: ${ 2{3+[4(2+1)]+1} }$.

Q: How do I use braces in a mathematical expression? A: To use braces in a mathematical expression, simply place the expression you want to evaluate last inside the braces. For example: ${ 2{1+[4(2+1)]+3} }$.

Q: What is the difference between parentheses, brackets, and braces? A: The main difference between parentheses, brackets, and braces is the order in which they are evaluated. Parentheses are evaluated first, followed by brackets, and finally braces.

Q: Can I use multiple grouping symbols in a mathematical expression? A: Yes, you can use multiple grouping symbols in a mathematical expression. For example: ${ 2{3+[4(2+1)]+1} }$.

Q: How do I evaluate a mathematical expression with multiple grouping symbols? A: To evaluate a mathematical expression with multiple grouping symbols, follow the order of operations and evaluate the expressions inside the grouping symbols from left to right.

Q: What are some common mistakes to avoid when using grouping symbols? A: Some common mistakes to avoid when using grouping symbols include:

  • Not following the order of operations
  • Not using the correct grouping symbols
  • Not evaluating expressions inside grouping symbols

Q: How can I practice using grouping symbols? A: You can practice using grouping symbols by working through practice problems, such as the ones provided in the previous article.

Q: What are some real-world applications of grouping symbols? A: Grouping symbols have many real-world applications, including:

  • Evaluating mathematical expressions in science and engineering
  • Writing computer code
  • Solving mathematical problems in finance and economics

Conclusion

Grouping symbols are an essential part of mathematical notation, and understanding how to use them is crucial for evaluating mathematical expressions correctly. By following the order of operations and using the correct grouping symbols, you can ensure that mathematical expressions are evaluated correctly. Practice problems and real-world applications are provided to help you understand grouping symbols and their importance.