Powers and Roots Order of Operations Practice Problems

Master the order of operations with powers and roots through step-by-step practice problems. Learn PEMDAS/BODMAS rules with square roots, cube roots, and exponents.

📚Practice Powers and Roots in Order of Operations
  • Solve expressions with square roots and exponents using PEMDAS
  • Calculate cube roots and higher powers in combined operations
  • Apply order of operations with parentheses, roots, and powers
  • Master multiplication and division of roots and exponents
  • Work through complex expressions with multiple operations
  • Build confidence solving real-world math problems with roots and powers

Understanding Order of Operations: Roots

Complete explanation with examples

As we have learned in previous lessons, when working with combined operations the order of the basic operations must be followed in order to get the correct result. However, before performing these the parentheses and then the roots and powers must first be solved.

Roots are very important in mathematical calculations. They are present in a variety of exercises ranging from algebraic problems for solving a second degree equation using the general formula, to geometric problems like determining the length of the hypotenuse of a right-angled triangle. Therefore, it is fundamental that we learn how to solve combined operations where this operation appears.

When we have simplified the root and power operations, we can continue solving the exercise according to the order of the basic operations: multiplications and divisions first, followed by additions and subtractions.

Let's revisit the order of the operations:

  1. Parentheses
  2. Powers and roots
  3. Multiplication and division
  4. Addition and subtraction
Detailed explanation of the BODMAS/PEMDAS rule highlighting 'Order' (Exponents) with symbols like √x and x², crucial for solving complex arithmetic expressions step by step.

Detailed explanation

Practice Order of Operations: Roots

Test your knowledge with 24 quizzes

\( 8-3^2:3= \)

Examples with solutions for Order of Operations: Roots

Step-by-step solutions included
Exercise #1

20−33:3= 20-3^3:3=

Step-by-Step Solution

First, compute the power: 33=27 3^3 = 27 .

Next, divide: 27÷3=9 27 \div 3 = 9 .

Finally, subtract: 20−9=11 20 - 9 = 11 .

Answer:

11 11

Exercise #2

15−42:2= 15-4^2:2=

Step-by-Step Solution

First, compute the power: 42=16 4^2 = 16 .

Next, divide: 16÷2=8 16 \div 2 = 8 .

Finally, subtract: 15−8=7 15 - 8 = 7 .

Answer:

7 7

Exercise #3

10−52:5= 10-5^2:5=

Step-by-Step Solution

First, compute the power: 52=25 5^2 = 25 .

Next, divide: 25÷5=5 25 \div 5 = 5 .

Finally, subtract: 10−5=5 10 - 5 = 5 .

Answer:

5 5

Exercise #4

8−16×3= 8 - \sqrt{16} \times 3 =

Step-by-Step Solution

First, evaluate the square root: 16=4\sqrt{16}=4.

Then, follow the order of operations (PEMDAS/BODMAS):

1. Multiplication: 4×3=124 \times 3 = 12

2. Subtraction: 8−12=−48 - 12 = -4

So, the correct answer is −4 -4 .

Answer:

−4 -4

Exercise #5

7+49−5= 7 + \sqrt{49} - 5 =

Step-by-Step Solution

First, evaluate the square root: 49=7\sqrt{49}=7.

Then, follow the order of operations (PEMDAS/BODMAS):

1. Addition: 7+7=147 + 7 = 14

2. Subtraction: 14−5=914 - 5 = 9

So, the correct answer is 9 9 .

Answer:

9 9

Frequently Asked Questions

Everything you need to know about Order of Operations: Roots

What is the correct order of operations with powers and roots?

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The order is: 1) Parentheses, 2) Powers and Roots (same priority level), 3) Multiplication and Division (left to right), 4) Addition and Subtraction (left to right). This follows the PEMDAS/BODMAS rule where powers and roots are solved before basic arithmetic operations.

Do you solve roots or powers first in math problems?

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Roots and powers have the same priority level in the order of operations. You can solve them in any order since they don't affect each other. Focus on simplifying all roots and powers before moving to multiplication, division, addition, and subtraction.

How do you solve order of operations problems with square roots?

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First solve operations in parentheses, then calculate all square roots and powers, next perform multiplication and division from left to right, and finally do addition and subtraction from left to right. For example: 5 + √49 + 4³ = 5 + 7 + 64 = 76.

What are common mistakes when solving expressions with roots and exponents?

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Common errors include: • Solving operations out of order • Forgetting to solve parentheses first • Not simplifying roots and powers before other operations • Mixing up square roots with regular multiplication • Not working left to right for same-priority operations

Can you multiply square roots in order of operations problems?

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Yes, you can multiply square roots, but follow the order of operations. First solve any parentheses, then calculate the individual roots, and finally multiply the results. For example: √9 × √4 = 3 × 2 = 6.

How do you handle cube roots in combined operations?

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Cube roots follow the same rules as square roots in the order of operations. Calculate the cube root value first (like ∛27 = 3), then proceed with other operations according to PEMDAS/BODMAS rules.

What grade level learns powers and roots in order of operations?

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Students typically learn basic order of operations with powers in grades 5-6, while roots are introduced in grades 7-8. Advanced combinations with multiple roots and powers are covered in pre-algebra and algebra courses.

Why do powers and roots come before multiplication in order of operations?

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Powers and roots are considered higher-level operations that must be simplified first to get accurate results. They have priority over basic arithmetic operations because they fundamentally change the values involved before other calculations can be performed correctly.

More Order of Operations: Roots Questions

Click on any question to see the complete solution with step-by-step explanations

Powers and Roots

Calculate (2/3)³: Finding the Cube of a Fraction Solve: 6 + Square Root of 64 - 4 Expression Evaluation Solve 18² - (100 + √9): Mixed Operations Expression Solve 2×(√36 + 9): Order of Operations with Square Root Solve 2×(3³ + √144): Order of Operations Practice

Continue Your Math Journey

Master Order of Operations: Roots first, then advance to these related topics that build on your fraction skills

Suggested Topics to Practice in Advance

The Order of Basic Operations: Addition, Subtraction, and Multiplication

Topics Learned in Later Sections

Order of Operations: ExponentsOrder of Operations - Exponents and RootsOrder of Operations with ParenthesesDivision and Fraction Bars (Vinculum)The Numbers 0 and 1 in OperationsNeutral Element (Identity Element)Order or Hierarchy of Operations with FractionsMultiplicative InverseSpecial cases (0 and 1, reciprocals, fraction line)The Order of Operations

Practice by Question Type

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Complete the missing numbers Exercises with fractions Identify the greater value Parentheses within parentheses Small Numbers Solving an exercise Solving the equation Solving the problem The perimeter of a square True / false Using fractions Using parentheses Using powers