Math Equation Solver

This versatile solver functions as a comprehensive order of operations or PEMDAS calculator. It accurately evaluates complex mathematical expressions by strictly following the PEMDAS algorithm, prioritizing operations in this exact sequence:

• Parenthesis, brackets, grouping
• Exponents, roots
• Multiplication, Division
• Addition, Subtraction

Directions for use

To use this PEMDAS solver, simply enter your mathematical equation using the following standard symbols:

• "+" Addition
• "-" Subtraction
• "*" Multiplication
• "/" Division
• "^" To the power of (E.g., 12^2 means 12 raised to the power of 2: 12² = 144. 49^(1/2) means 49 raised to the power of 1/2: 49¹/² = 7).
• "root"(x[n])
• You can use (), {}, [] for brackets and grouping.

Copying equations from other sources

You can easily copy and paste expressions from external sources directly into this math equation calculator. In most cases, the calculator will automatically process the equation even if the source text uses non-standard symbols, such as × instead of * or ÷ instead of /. However, in some rare instances, you may need to manually replace unrecognized characters with the standard operators listed above.

Working with fractions

This order of operations calculator fully supports fractions. Use the forward slash / as the fraction bar and enclose the entire fraction in brackets to ensure precise calculations. If you omit the brackets, the fractional division will be processed according to the strict PEMDAS order of operations. For example, enter 25^(1/2) to calculate 25 to the power of 1/2: 25^(1/2) = 5. If you enter 25^1/2 without the grouping brackets, the calculator will evaluate it as (25^1)/2 = 25/2 = 12.5, adhering strictly to the PEMDAS rule.

PEMDAS order of operations

When a mathematical expression contains only one operation, the answer is usually straightforward. For example, 12 + 4 = 16.

However, how do you evaluate a more complex expression like this: 3 × 4 – 4? Which operation should take precedence? If you perform the multiplication first, you get 3 × 4 – 4 = 12 – 4 = 8. But if you calculate the subtraction first, the result changes entirely: 3 × 4 – 4 = 3 × 0 = 0.

To eliminate this ambiguity, mathematicians assign strict priorities to all mathematical operations and ALWAYS perform them in a standardized sequence. This universal rule is commonly described by the PEMDAS acronym, where P stands for parenthesis (or brackets, or grouping), E means exponents (and roots), M means multiplication, D means division, A means addition, and S means subtraction.

Keep in mind that different countries use varying acronyms, but they all describe the exact same order of operations. For instance, BEDMAS stands for Brackets, Exponents, Division, Multiplication, Addition, Subtraction; GEMDAS is an acronym for Grouping, Exponents, Multiplication, Division, Addition, Subtraction; and BODMAS means Brackets, Order, Division, Multiplication, Addition, Subtraction.

The order of multiplication and division

Under the PEMDAS algorithm, multiplication and division share equivalent priority. This means they are evaluated sequentially from left to right as they appear in the equation (unless one is enclosed in brackets). For example, in the expression 12 / 2 × 3, you will first perform the division 12 / 2 to get 6, and then multiply 6 by 3 to arrive at a final answer of 18.

This equivalent priority explains why in some acronyms, M (Multiplication) comes before D (Division) like in PEMDAS, while in others, D precedes M like in BODMAS.

The order of addition and subtraction

Addition and subtraction also hold equivalent priority levels. These operations are executed as soon as they appear in the mathematical expression, reading from left to right. For example, in the equation 10 – 7 + 3, you must first perform the subtraction 10 – 7 = 3, followed by the addition 3 + 3 = 6. Ultimately, 10 – 7 + 3 = 6.

The order of roots and exponents

As explained above, multiplication, division, addition, and subtraction are all left-associative operations—meaning they are resolved from left to right. Conversely, roots and exponents are right-associative operations, meaning they are evaluated from right to left.

For example, let’s solve the following expression: 2^3^1^2 or $2^{3^{1^{2}}}$.

Since an exponent is a right-associative operation, we begin calculating on the right side.

We first calculate 1^2=1, then 3^1=3, and finally 2^3=8. This unique sequence is sometimes referred to as a “top-down order,” as you start with the top-most exponent and work your way “down” the equation.

The expression can be re-written as follows:

2^3^1^2 = 2^(3^(1^2)) = 2^(3^1) = 2^3 = 8

$$2^{3^{1^{2}}} = 2^{3^{1}} = 2^{3} = 8$$

Multiple brackets

When evaluating an expression that features multiple sets of brackets, the calculation always starts with the innermost bracket and systematically works its way outward to the outer brackets. Note that if the expression inside a bracket contains several different operations, they must still be resolved according to the strict PEMDAS order of operations.

Real life example

At first glance, the order of operations might seem like a strictly theoretical mathematical concept. However, we actively use it in our everyday lives without even realizing it!

Imagine you are ordering pizzas with a group of friends. Let’s say you order a Margherita pizza for $15, a Quattro Formaggi for $16.50, and a Neapolitan pizza for $14.50. You are a group of 8 people, and you need to calculate how much each person owes. To find the exact split, you are essentially solving the following mathematical expression using the PEMDAS algorithm:

(15 + 16.50 + 14.50)/8 = (31.50 + 14.50)/8 = (46)/8 = 46/8 = 5.75

Each person will need to pay $5.75.

Remembering the acronym

Many fun mnemonic phrases are used to help students remember the PEMDAS acronym, with the most famous being “Please Excuse My Dear Aunt Sally.” By taking the first letter of each word in the phrase, you easily spell out PEMDAS. You can use this classic phrase or get creative and invent your own—for example, “Purple Elves Make Dull Affordable Sausages!”