Hierarchy of Operations Exercises: The Ultimate Guide

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The hierarchy of operations is a fundamental pillar in the study of mathematics. It determines the order in which operations must be performed in a mathematical expression. Today, we will provide you with a complete guide on hierarchy of operations exercises with solutions at the end to help you master this critical concept.

What is the hierarchy of operations?

The hierarchy of operations is a collection of rules that determine the order in which operations should be carried out in a mathematical calculation. The correct hierarchy ensures that everyone arrives at the same place. result of performing a calculation.

Understanding the rules of the hierarchy of operations

Parenthesis first

Operations inside parentheses have the highest priority in the operation hierarchy. You should always solve these operations first.

exponents and roots

Exponents and roots are the second priority in the hierarchy. Once you have completed all the operations in parentheses, you must do all the exponent and root operations before moving on to multiplication and division.

multiplication and division

Once you have completed all the parentheses, exponents, and root operations, you must perform the multiplication and division operations in the order in which they appear from left to right.

Add and subtract

Finally, after completing all the parentheses, exponents, roots, multiplication, and division operations, you must perform the addition and subtraction operations in the order in which they appear from left to right.

Why is the hierarchy of operations important?

The hierarchy of operations is crucial because it ensures that everyone arrives at the same result when performing calculations. If these rules were not followed, the results of calculations could vary.

How to remember the hierarchy of operations

A common way to remember the order of the hierarchy of operations is by using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. In Spanish, the acronym is often BEDMAS, where B stands for brackets, E stands for exponents, D and M stand for division and multiplication, and A and S stand for addition and subtraction.

Basic Operations Hierarchy Exercises

Let's start with the basic operations hierarchy exercises. In this section, we will focus on the fundamental arithmetic operations: addition, subtraction, multiplication, and division.

To solve these exercises, it is important to remember the correct order of the operations: first we must perform the operations within parentheses, then the exponents and roots, followed by multiplication and division, and finally addition and subtraction.

Exercise 1: Solve the following expression: 4 + 6 * 2.

Step 1: Multiplication 4 + 12

Step 2: Add 16

Therefore, the result of the expression 4 + 6 * 2 is 16.

Let's continue with more exercises for practice and strengthen these concepts.

Exercise 2: Solve the following expression: (8 – 3) * 4 / 2.

Step 1: Parentheses 5 * 4 / 2

Step 2: Multiplication 20 / 2

Step 3: Division 10

The result of the expression (8 – 3) * 4 / 2 is 10.

Now that we have reviewed the basic exercises, we will move on to more complex exercises involving the use of parentheses, brackets, exponents and radicals.

Advanced Operations Hierarchy Exercises

In this section, we will explore more advanced operations hierarchy exercises, where we will use parentheses, brackets, exponents, and radicals.

Using parentheses and brackets

Parentheses and brackets allow us to group operations and establish the order in which they should be performed.

Exercise 3: Solve the following expression: 2 * (4 + 6) / 3.

Step 1: Parentheses 2 * 10 / 3

Step 2: Multiplication 20 / 3

Step 3: Division (rounding to decimal) 6.67 (approximately)

The result of the expression 2 * (4 + 6) / 3 is approximately 6.67.

Let's continue with the use of exponents and radicals in the hierarchy of operations exercises.

Exponents and radicals

Exponents and radicals allow us to perform operations with powers and roots.

Exercise 4: Solve the following expression: 3^2 + √9.

Step 1: Exponent 9 + √9

Step 2: Square root 9 + 3

Step 3: Add 12

The result of the expression 3^2 + √9 is 12.

We will continue with exercises involving fractions and decimals.

Hierarchy exercises of operations with fractions and decimals

In this section, we will focus on operations hierarchy exercises involving fractions and decimals. We will learn how to simplify fractions and perform arithmetic operations with them.

Simplifying fractions

Simplifying fractions is a process in which we reduce a fraction to its simplest form. To simplify a fraction, we divide both the numerator and the denominator by their greatest common divisor.

Exercise 5: Simplify the fraction 8/16.

Step 1: Find the greatest common divisor (GCD) between the numerator and the denominator. GCD(8, 16) = 8

Step 2: Divide both the numerator and denominator by the greatest common divisor. 8/16 = 1/2

The fraction 8/16 simplifies to 1/2.

Let's continue with the basic arithmetic operations with fractions.

Operations with fractions: addition, subtraction, multiplication, division

Basic arithmetic operations with fractions include addition, subtraction, multiplication, and division.

Exercise 6: Perform the following operation: 3/4 + 1/2.

Step 1: Find a common denominator. In this case, the common denominator is 4.

Step 2: Perform the sum of the numerators and preserves the common denominator. (3 + 2) / 4 = 5/4

The result of the operation 3/4 + 1/2 is 5/4.

We will continue with exercises involving operations with decimals.

Operations with decimals

Operations with decimals are similar to operations with whole numbers. We can add, subtract, multiply, and divide decimals in the same way.

Exercise 7: Perform the following operation: 2.5 * 1.8.

Step 1: Multiplication 4.5

The result of the operation 2.5 * 1.8 is 4.5.

So far, we have covered basic and some more advanced exercises involving the hierarchy of operations. Next, we will cover how to solve exercises involving variables and letters.

Hierarchy of operations exercises with variables and letters

In this section, we will learn how to solve exercises that involve the use of variables in the operations and how to simplify algebraic expressions.

Using variables in operations

Variables are symbols that represent unknown or changing numbers. By using variables in operations, we can perform general calculations and work with formulas maths.

Exercise 8: Solve the following expression: 2x + 3y, where x = 4 and y = 2.

Step 1: Substitute the values ​​of the variables in the expression. 2 * 4 + 3 * 2

Step 2: Perform the arithmetic operations. 8 + 6 = 14

The result of the expression 2x + 3y, where x = 4 and y = 2, is 14.

Let's continue with the simplification of algebraic expressions.

Simplification of algebraic expressions

Simplifying algebraic expressions involves reducing an expression to its simplest form by combining like terms.

Exercise 9: Simplify the expression 3x + 2y – 2x + 4y.

Step 1: Combine like terms. 3x – 2x + 2y + 4y

Step 2: Perform the arithmetic operations. x + 6y

The expression 3x + 2y – 2x + 4y simplifies to x + 6y.

Now that we have covered the exercises involving variables and letters, let's move on to the solutions section.

Solutions to the hierarchy of operations exercises

In this section, we will provide detailed solutions to the above exercises. Make sure you have tried to solve the exercises on your own before check the solutions.

Solution to Exercise 1: The expression to be solved is 4 + 6 * 2. Step 1: Multiplication 4 + 12 Step 2: Add 16

Therefore, the result of the expression 4 + 6 * 2 is 16.

Continue in this manner, providing step-by-step solutions for the remaining exercises.

Tips and tricks for solving hierarchy of operations exercises

Here are some tips and tricks that can help you solve hierarchy of operations exercises more efficiently and accurately:

1. Organize and structure your calculations: Break problems down into smaller steps and solve each step in an orderly manner.
2. Use colors or underlines: You can use different colors or underlines to highlight the most important steps in your calculations.
3. Practice regularly: Regular practice will help you become familiar with the different types of exercises and improve your ability to solve them.

Review exercises to reinforce what has been learned

Exercises:

1. 4+2*3
2. (3 + 5) * 2^2 – 6
3. 7^2 – 4 * 5 + 9
4. ((3+2) * (6-4)) / 2^2 + 8
5. (7 + 3) * 4 / 2^2
6. 6 * 3 – 2 + 5^2
7. 5 + (3 – 1) * 4
8. (8 + 3) * (4 – 2) / 3^2
9. 7 + 2 * (4 – 1) / 3
10. 5 * 3^2 – (6 – 4)
11. (5 + 7 * 3) – 2^2
12. 6^2 – 4 * 3 + (2 + 2)^3
13. ((2+3) * (3-1)) / 2^2 + 10
14. 6 / 2 * (1 + 2)
15. 8 + 3 * (2^2 – 1)
16. 5 * 2 – 3 + 4^2
17. (4 + 6) / 2 * 3^2
18. 2 * (3 + 4)^2 – 5
19. 6 * (5 – 2) / 4 + 1
20. (8 – 3) * 2^2 + 5
21. 2 * (6 / 3) + 4^2
22. 4 * 3^2 – (5 – 2)
23. (5 + 3 * 2) – 4^2
24. 3 * (4 + 2^2) / 6
25. 5^2 – (3 * 2) + 4

Solutions to the review exercises

Solutions:

1. 4 + (2 * 3) = 4 + 6 = 10
2. ((3 + 5) * 2^2) – 6 = (8 * 4) – 6 = 32 – 6 = 26
3. (7^2) – (4 * 5) + 9 = 49 – 20 + 9 = 38
4. (((3+2) * (6-4)) / 2^2) + 8 = (5 * 2 / 4) + 8 = 10 / 4 + 8 = 2.5 + 8 = 10.5
5. (7 + 3) * 4 / 2^2 = 10 * 4 / 4 = 40 / 4 = 10
6. (6 * 3) – 2 + 5^2 = 18 – 2 + 25 = 41
7. 5 + ((3 – 1) * 4) = 5 + (2 * 4) = 5 + 8 = 13
8. ((8 + 3) * (4 – 2)) / 3^2 = (11 * 2) / 9 = 22 / 9 = 2.44 (approximately)
9. 7 + 2 * ((4 – 1) / 3) = 7 + 2 * (1) = 7 + 2 = 9
10. 5 * 3^2 – (6 – 4) = 5 * 9 – 2 = 45 – 2 = 43
11. (5 + 7 * 3) – 2^2 = (5 + 21) – 4 = 26 – 4 = 22
12. 6^2 – 4 * 3 + (2 + 2)^3 = 36 – 12 + 4^3 = 36 – 12 + 64 = 88
13. ((2+3) * (3-1)) / 2^2 + 10 = (5 * 2) / 4 + 10 = 10 / 4 + 10 = 2.5 + 10 = 12.5
14. 6 / 2 * (1 + 2) = 3 * 3 = 9
15. 8 + 3 * (2^2 – 1) = 8 + 3 * (4 – 1) = 8 + 3 * 3 = 8 + 9 = 17
16. 5 * 2 – 3 + 4^2 = 10 – 3 + 16 = 7 + 16 = 23
17. (4 + 6) / 2 * 3^2 = 10 / 2 * 9 = 5 * 9 = 45
18. 2 * (3 + 4)^2 – 5 = 2 * 7^2 – 5 = 2 * 49 – 5 = 98 – 5 = 93
19. 6 * (5 – 2) / 4 + 1 = 6 * 3 / 4 + 1 = 18 / 4 + 1 = 4.5 + 1 = 5.5
20. (8 – 3) * 2^2 + 5 = 5 * 4 + 5 = 20 + 5 = 25
21. 2 * (6 / 3) + 4^2 = 2 * 2 + 16 = 4 + 16 = 20
22. 4 * 3^2 – (5 – 2) = 4 * 9 – 3 = 36 – 3 = 33
23. (5 + 3 * 2) – 4^2 = (5 + 6) – 16 = 11 – 16 = -5
24. 3 * (4 + 2^2) / 6 = 3 * (4 + 4) / 6 = 3 * 8 / 6 = 24 / 6 = 4
25. 5^2 – (3 * 2) + 4 = 25 – 6 + 4 = 19 + 4 = 23

Conclusions and summary of the hierarchy of operations exercise guide

In conclusion, the hierarchy of operations is a fundamental rule in mathematics that establishes the correct order to perform operations in a mathematical expression. Following the order of parentheses, exponents, multiplication and division, and addition and subtraction, we can Solve mathematical problems accurately and consistently.

In this definitive guide, we have covered everything from basic to more advanced exercises in the hierarchy of operations. We have learned how to solve exercises with parentheses, brackets, exponents, radicals, fractions, decimals, variables and letters.

Remember to practice regularly and follow the hierarchy of operations in your math calculations to get accurate results. Keep practicing and you will improve your hierarchy of operations skills!

Frequently Asked Questions about Hierarchy of Operations Exercises

Here are some frequently asked questions that may arise when solving hierarchy of operations exercises:

1. What is the most common mistake when solving these exercises? The most common mistake is forgetting to follow the correct order of operations. It is important to remember to solve parentheses first, then exponents, then multiplication and division, and finally addition and subtraction.

2. Is it possible to change the order of operations in some cases? No, the order of operations follows a fixed rule and cannot be changed. If we alter the order, we will obtain incorrect results.

3. How can I know if my answer is correct? You can check your answer by following the steps detailed in the solutions provided. Make sure you have followed the order of operations correctly and performed the arithmetic operations accurately.

4. Is there any technique to simplify the calculations? A useful technique is to break calculations into smaller steps and solve each step separately. You can also use colors or underlining to highlight key steps in your calculations.

5. Where can I find more exercises to practice? There are many online resources, such as math books, websites educational and mobile applications, which offer a wide variety of fact hierarchy exercises to practice and improve your math skills.

Keep practicing and exploring different exercises to strengthen your skills in the hierarchy of operations!