This solver can be used as an order of operations or PEMDAS calculator. It solves mathematical problems following the PEMDAS algorithm, prioritizing the operations as follows:
- Parenthesis, brackets, grouping
- Exponents, roots
- Multiplication, Division
- Addition, Subtraction
Directions for use
To use this PEMDAS solver, enter the given equation using the following 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 copy and paste equations from other sources into this equation calculator. The calculator will usually work even if the source file uses different symbols for operations, for example, × instead of * or ÷ instead of /. In some cases, however, you will have to replace the different symbols with the ones recognized by this calculator.
Working with fractions
This calculator also works with fractions. Use the fraction bar / to enter a fraction, and enclose the given fraction in brackets. Otherwise, the fractional division will be performed according to the PEMDAS order of operations. For example, enter 25^(1/2) to have 25 to the power of 1/2: 25^(1/2) = 5. If you enter 25^1/2, you will get 12.5 as the answer since the calculator will interpret 25^1/2 as (25^1)/2 = 25/2 = 12.5, following the PEMDAS order.
PEMDAS order of operations
If you have only one operation in a mathematical expression, the answer is usually clear. For example, 12 + 4 = 16.
However, what do you do with an expression like this: 3 × 4 – 4? Which operation should you perform first? If you do the multiplication first, you will get 3 × 4 – 4 = 12 – 4 = 8. But if you do the subtraction first, you will get a different answer: 3 × 4 – 4 = 3 × 0 = 0.
To solve this problem, mathematicians assign priorities to all operations and ALWAYS perform them in a specific order. This order is described by the PEMDAS acronym, where P stands for parenthesis (or brackets, or grouping), E – means exponents (and roots), M – means multiplication, D – division, A – addition, S – subtraction.
Note that different countries use different acronyms, but they all describe the same order of operations. For example, BEDMAS stands for Brackets, Exponents, Division, Multiplication, Addition, Subtraction; GEMDAS is an acronym for Grouping, Exponents, Multiplication, Division, Addition, Subtraction; BODMAS means Brackets, Order, Division, Multiplication, Addition, Subtraction.
The order of multiplication and division
In the PEMDAS algorithm, multiplication and division are equivalent priority operations, meaning they are simply performed from left to right (unless one of them is in brackets). For example, in the expression 12 / 2 × 3 you will first perform the division 12 / 2 to get 6, then multiply 6 by 3 to get 18.
That’s why in some acronyms M – Multiplication stands before D – Division (PEMDAS), while in others, D stands before M (BODMAS).
The order of addition and subtraction
Addition and subtraction also have equivalent priority. These operations are performed as soon as they occur in the expression, from left to right. For example, in the expression 10 – 7 + 3, you first need to perform subtraction 10 – 7 = 3, and then addition 3 + 3 = 6. 10 – 7 + 3 = 6.
The order of roots and exponents
As described above, the operations of multiplication and division, as well as addition, and subtraction operations, are performed from left to right. These operations are called left-associative. On the other hand, roots and exponents are right-associative, meaning they are performed from right to left.
For example, let’s solve the following expression: 2^3^1^2 or \$2^{3^{1^{2}}}\$.
Exponent is a right-associative operation, so we begin the solution on the right side.
We first calculate 1^2=1, then 3^1=3, and finally 2^3=8. This order is sometimes described as “top to down order,” as you start with the top-most exponent and make your way “down.”
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 an expression has multiple brackets, the solution starts with the innermost bracket and proceeds to the outer brackets. Note that if the expression inside the brackets contains several operations, they are still performed following the PEMDAS order.
Real life example
At first glance, the order of operations seems to be a strictly mathematical concept. However, we very often use it in everyday life without even noticing! For example, imagine you are ordering pizzas with a group of friends. Let’s say you order one pizza Margherita for $15, one Pizza quattro formaggi for $16.50, and one Neapolitan pizza for $14.50. You are a group of 8 people, and you need to calculate how much each of you has to pay. To do that, you will essentially solve the following expression using the PEMDAS algorithm:
(15 + 16.50 + 14.50)/8 = (31.50 + 14.50)/8 = (46)/8 = 46/8 = 5.75
Each of you will have to pay $5.75.
Remembering the acronym
Many phrases are used to remember the PEMDAS acronym, the most common one being “Please Excuse My Dear Aunt Sally.” Taking the first letter of each of the words, you will get PEMDAS. Use this phrase, or come up with your own, for example, “Purple Elves Make Dull Affordable Sausages!”
