Use Brackets To Evaluate: ${ (30 \div 6) + (5 \times 2) - (4 \div 2) }$

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Introduction to Order of Operations

When evaluating mathematical expressions, it's essential to follow the order of operations, which is a set of rules that dictate the order in which operations should be performed. 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 multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

However, in this case, we're dealing with brackets, which serve a similar purpose to parentheses. Brackets are used to group expressions and indicate the order in which operations should be performed.

Evaluating the Expression Inside Brackets

Let's start by evaluating the expression inside the brackets:

$(30 \div 6) + (5 \times 2) - (4 \div 2)$

We can break this down into three separate expressions:

1. $(30 \div 6)$
2. $(5 \times 2)$
3. $(4 \div 2)$

Evaluating the First Expression

The first expression is $(30 \div 6)$. To evaluate this expression, we need to follow the order of operations. Since there are no parentheses, exponents, or multiplication and division operations, we can simply divide 30 by 6:

$(30 \div 6) = 5$

Evaluating the Second Expression

The second expression is $(5 \times 2)$. To evaluate this expression, we need to follow the order of operations. Since there are no parentheses, exponents, or division operations, we can simply multiply 5 by 2:

$(5 \times 2) = 10$

Evaluating the Third Expression

The third expression is $(4 \div 2)$. To evaluate this expression, we need to follow the order of operations. Since there are no parentheses, exponents, or multiplication operations, we can simply divide 4 by 2:

$(4 \div 2) = 2$

Combining the Results

Now that we've evaluated each expression inside the brackets, we can combine the results:

$5 + 10 - 2$

Evaluating the Final Expression

To evaluate the final expression, we need to follow the order of operations. Since there are no parentheses, exponents, or multiplication and division operations, we can simply add and subtract the numbers from left to right:

$5 + 10 = 15$ $15 - 2 = 13$

Therefore, the final result is:

$(30 \div 6) + (5 \times 2) - (4 \div 2) = 13$

Conclusion

In conclusion, when evaluating mathematical expressions with brackets, it's essential to follow the order of operations and evaluate the expressions inside the brackets first. By breaking down the expression into smaller parts and following the order of operations, we can arrive at the final result.

Frequently Asked Questions

• Q: What is the order of operations? A: The order of operations is a set of rules that dictate the order in which operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations.
• Q: What is the difference between parentheses and brackets? A: Parentheses and brackets are both used to group expressions and indicate the order in which operations should be performed. However, parentheses are used more frequently in mathematical expressions.
• Q: How do I evaluate an expression with brackets? A: To evaluate an expression with brackets, you need to follow the order of operations and evaluate the expressions inside the brackets first.

Additional Resources

• Khan Academy: Order of Operations
• Mathway: Order of Operations
• Wolfram Alpha: Order of Operations

Final Thoughts

Evaluating mathematical expressions with brackets requires a clear understanding of the order of operations and the ability to break down complex expressions into smaller parts. By following the order of operations and evaluating the expressions inside the brackets first, we can arrive at the final result.

Introduction

Evaluating mathematical expressions with brackets can be a challenging task, especially for those who are new to mathematics. In this article, we will answer some of the most frequently asked questions about evaluating mathematical expressions with brackets.

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which operations should be performed. 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 multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between parentheses and brackets?

A: Parentheses and brackets are both used to group expressions and indicate the order in which operations should be performed. However, parentheses are used more frequently in mathematical expressions. Brackets are often used to group expressions that are not enclosed in parentheses.

Q: How do I evaluate an expression with brackets?

A: To evaluate an expression with brackets, you need to follow the order of operations and evaluate the expressions inside the brackets first. Here's a step-by-step guide:

1. Evaluate any expressions inside the brackets.
2. Evaluate any expressions inside the parentheses.
3. Evaluate any exponential expressions.
4. Evaluate any multiplication and division operations from left to right.
5. Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between a bracket and a parenthesis?

A: A bracket is a symbol that is used to group expressions, while a parenthesis is a symbol that is used to group expressions and indicate the order in which operations should be performed. Brackets are often used to group expressions that are not enclosed in parentheses.

Q: Can I use both brackets and parentheses in the same expression?

A: Yes, you can use both brackets and parentheses in the same expression. However, it's essential to follow the order of operations and evaluate the expressions inside the brackets and parentheses first.

Q: How do I know which operation to perform first?

A: To determine which operation to perform first, you need to follow the order of operations. If there are any expressions inside the brackets or parentheses, evaluate those first. If there are any exponential expressions, evaluate those next. Finally, evaluate any multiplication and division operations from left to right, followed by any addition and subtraction operations from left to right.

Q: Can I use brackets to group expressions that are not enclosed in parentheses?

A: Yes, you can use brackets to group expressions that are not enclosed in parentheses. However, it's essential to follow the order of operations and evaluate the expressions inside the brackets first.

Q: How do I evaluate an expression with multiple brackets?

A: To evaluate an expression with multiple brackets, you need to follow the order of operations and evaluate the expressions inside the brackets from left to right. Here's a step-by-step guide:

1. Evaluate any expressions inside the innermost brackets.
2. Evaluate any expressions inside the next set of brackets.
3. Continue evaluating expressions inside brackets until you have evaluated all of them.
4. Finally, evaluate any addition and subtraction operations from left to right.

Q: Can I use brackets to group expressions that are enclosed in parentheses?

A: Yes, you can use brackets to group expressions that are enclosed in parentheses. However, it's essential to follow the order of operations and evaluate the expressions inside the parentheses first.

Q: How do I know if an expression is enclosed in parentheses or brackets?

A: To determine if an expression is enclosed in parentheses or brackets, look for the symbols. If the expression is enclosed in parentheses, it will have a pair of parentheses (()) at the beginning and end of the expression. If the expression is enclosed in brackets, it will have a pair of brackets [[]] at the beginning and end of the expression.

Q: Can I use brackets to group expressions that are not enclosed in parentheses or brackets?

A: Yes, you can use brackets to group expressions that are not enclosed in parentheses or brackets. However, it's essential to follow the order of operations and evaluate the expressions inside the brackets first.

Q: How do I evaluate an expression with multiple parentheses and brackets?

A: To evaluate an expression with multiple parentheses and brackets, you need to follow the order of operations and evaluate the expressions inside the parentheses and brackets from left to right. Here's a step-by-step guide:

1. Evaluate any expressions inside the innermost parentheses.
2. Evaluate any expressions inside the next set of parentheses.
3. Evaluate any expressions inside the innermost brackets.
4. Evaluate any expressions inside the next set of brackets.
5. Continue evaluating expressions inside parentheses and brackets until you have evaluated all of them.
6. Finally, evaluate any addition and subtraction operations from left to right.

Conclusion

Evaluating mathematical expressions with brackets can be a challenging task, but by following the order of operations and understanding the difference between parentheses and brackets, you can arrive at the correct solution. Remember to evaluate expressions inside the brackets and parentheses first, and then follow the order of operations to evaluate any addition and subtraction operations from left to right.