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Order of Operations Calculator with Exponents and Square Roots

Reviewed by Calculator Editorial Team

This order of operations calculator solves complex mathematical expressions involving parentheses, exponents, square roots, multiplication, division, addition, and subtraction. Whether you're a student, teacher, or professional, this tool helps you solve equations accurately and efficiently.

How to Use This Calculator

Enter your mathematical expression in the input field below. The calculator will evaluate the expression according to the standard order of operations (PEMDAS/BODMAS rules).

Tip: Use parentheses to group operations and ensure the correct order of evaluation. For example, (3 + 4) × 2 evaluates to 14, while 3 + 4 × 2 evaluates to 11.

Input Rules

  • Use standard mathematical operators: + (addition), - (subtraction), × (multiplication), ÷ (division), ^ (exponentiation), √ (square root)
  • Parentheses can be used to override the default order of operations
  • Enter numbers directly without commas or spaces
  • For square roots, use the √ symbol followed by the number (e.g., √9 for the square root of 9)

Example Inputs

  • 3 + 4 × 2
  • (3 + 4) × 2
  • 2^3 + √16
  • 5 × (2 + 3) ÷ 4

Understanding Order of Operations

The order of operations is a set of rules that determines the sequence in which mathematical operations should be performed in an expression. The standard order is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction).

Order of Operations Rules:

  1. Parentheses/Brackets
  2. Exponents/Orders
  3. Multiplication and Division (from left to right)
  4. Addition and Subtraction (from left to right)

When evaluating expressions, always follow these rules to ensure accurate results. The calculator automatically applies these rules when evaluating your expressions.

Working with Exponents

Exponents indicate repeated multiplication. For example, 2^3 means 2 multiplied by itself 3 times (2 × 2 × 2 = 8).

Exponent Rules:

  • a^b means a multiplied by itself b times
  • a^0 = 1 for any a ≠ 0
  • a^1 = a
  • a^(b+c) = a^b × a^c

When exponents appear in an expression, they are evaluated before multiplication and division according to the order of operations.

Calculating Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, √9 = 3 because 3 × 3 = 9.

Square Root Properties:

  • √(a × b) = √a × √b
  • √(a^2) = |a|
  • √(a + b) ≠ √a + √b (in general)

The calculator handles square roots by evaluating them according to the order of operations rules. For negative numbers, the calculator will return a complex number result.

Worked Examples

Example 1: Basic Order of Operations

Expression: 3 + 4 × 2

Solution:

  1. Multiplication first: 4 × 2 = 8
  2. Then addition: 3 + 8 = 11

Final result: 11

Example 2: Parentheses Override

Expression: (3 + 4) × 2

Solution:

  1. Parentheses first: 3 + 4 = 7
  2. Then multiplication: 7 × 2 = 14

Final result: 14

Example 3: Exponents and Square Roots

Expression: 2^3 + √16

Solution:

  1. Exponent first: 2^3 = 8
  2. Then square root: √16 = 4
  3. Finally addition: 8 + 4 = 12

Final result: 12

Frequently Asked Questions

What is the correct order of operations?
The correct order is Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right).
How do I handle negative numbers in square roots?
The calculator will return a complex number result for negative numbers under the square root. For example, √-1 = i (where i is the imaginary unit).
Can I use decimal numbers in the calculator?
Yes, you can enter decimal numbers directly in the input field. The calculator will handle them according to the standard order of operations.
What happens if I enter an invalid expression?
The calculator will display an error message if the expression is invalid or cannot be evaluated. Please check your input for syntax errors.
Is there a way to save my calculations?
Currently, the calculator does not save calculations. You can manually note down important results or use your browser's print function to save the page.