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 27 quizzes

\( 6 - 3 + 5 \times 2^2 = \)

Examples with solutions for Order of Operations: Roots

Step-by-step solutions included
Exercise #1

10:2−22= 10:2-2^2=

Step-by-Step Solution

The given mathematical expression is 10:2−22 10:2-2^2 .

According to the order of operations (often remembered by the acronym PEMDAS/BODMAS), we perform calculations in the following sequence:

  • Parentheses/Brackets
  • Exponents/Orders (i.e., powers and roots)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

In this expression, there are no parentheses, but there is an exponent: 222^2. We calculate the exponent first:

22=42^2 = 4

Substituting back into the expression, we have:

10:2−4 10:2-4

Next, we perform the division from left to right. Here, ":" is interpreted as division:

10÷2=5 10 \div 2 = 5

Now, substitute this back into the expression:

5−4 5 - 4

The final step is to perform the subtraction:

5−4=1 5 - 4 = 1

Therefore, the answer is 1 1 .

Answer:

1

Video Solution
Exercise #2

3×3+32= ? 3\times3+3^2=\text{ ?}

Step-by-Step Solution

First we need to remind ourselves of the order of operations:

  1. Parentheses

  2. Exponents and Roots

  3. Multiplication and Division

  4. Addition and Subtraction

There are no parentheses in this problem, therefore we will start with exponents:

3 * 3 + 3² =

3 * 3 + 9 =

Let's continue to the next step—multiplication operations:

3 * 3 + 9 =

9 + 9 =

Finally, we are left with a simple addition exercise:

9 + 9 = 18

Answer:

18

Video Solution
Exercise #3

8−32:3= 8-3^2:3=

Step-by-Step Solution

Let's solve the expression step by step using the order of operations, often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

The given expression is: 8−32:3= 8-3^2:3=

Step 1: Evaluate Exponents
The expression has an exponent, which we need to evaluate first. The exponent is 323^2.
Calculate 323^2 which equals 99.
Now the expression becomes: 8−9:3 8 - 9 : 3

Step 2: Division
Next, perform the division operation. Here we divide 99 by 33.
Calculate 9:39 : 3 which equals 33.
Now the expression becomes: 8−3 8 - 3

Step 3: Subtraction
Finally, perform the subtraction.
Calculate 8−38 - 3 which equals 55.

Therefore, the solution to the expression 8−32:38-3^2:3 is 55.

Answer:

5 5

Video Solution
Exercise #4

4+2+52= 4+2+5^2=

Step-by-Step Solution

To solve the expression 4+2+52 4 + 2 + 5^2 , we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

  • Step 1: Calculate Exponents
    In the expression we have an exponent: 525^2. This means 5 is raised to the power of 2. We calculate this first:
    52=255^2 = 25.

  • Step 2: Perform Addition
    Now, substitute the calculated value back into the expression:
    4+2+254 + 2 + 25.
    Perform the additions from left to right:
    4+2=64 + 2 = 6
    Finally add the result to 25:
    6+25=316 + 25 = 31.

Therefore, the final answer is 3131.

Answer:

31

Video Solution
Exercise #5

4+22= 4+2^2=

Step-by-Step Solution

To solve the expression 4+22 4 + 2^2 , follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Let's break down the expression:

  • Step 1: Identify any exponents.
    The expression contains an exponent: 22 2^2 . To evaluate this, multiply 2 by itself: 2×2 2 \times 2 , which equals 4.
    So, 22=4 2^2 = 4 .
  • Step 2: Perform addition.
    Now, substitute the result back into the original expression:
    4+4 4 + 4 .
    Add these numbers together: 4 + 4 equals 8.

Therefore, the answer to the expression 4+22 4 + 2^2 is 8.

Answer:

8

Video Solution

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 18² - (100 + √9): Mixed Operations Expression Solve 2×(√36 + 9): Order of Operations with Square Root Solve 2×(3³ + √144): Order of Operations Practice Evaluate (4² + 3) × √9: 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

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