follow the Order of Operations
Step-by-step explanation:
An algebraic expression cannot be evaluated unless all of its variables have been replaced by numerical values. (It can be simplified, but not evaluated if it contains variables.)
A collection of numbers and math symbols is interpreted according to the Order of Operations. This order reflects a precedence of operations that is generally agreed or understood to be applied to algebraic expressions. Operations with the highest precedence are performed first. Operations with equal precedence are generally performed in order, left to right. (There are exceptions.) Parentheses or other grouping symbols are used to modify the order of operations as may be necessary.
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Here is a description of the most often seen operations in an algebraic expression, in order of precedence (highest to lowest).
1. Parentheses or Brackets -- any expression enclosed in parentheses or brackets is evaluated first. Evaluation is according to the order of operations. That means that if parentheses are nested, expressions in the innermost parentheses are evaluated first.
2. Exponents or Indices -- Expressions with exponents are evaluated next. In this context, roots are fractional exponents. If exponents are nested, they are applied right to left:
3^2^4 = 3^(2^4) = 3^16 = 43,046,721 . . . for example
Parentheses modify this order, so ...
(3^2)^4 = 9^4 = 6,561
The exponent is taken to be the first number immediately following the exponentiation symbol, so ...
9^1/2 = (9^1)/2 = 9/2 = 4.5
Again, parentheses alter this order, so ...
9^(1/2) = √9 = 3
3. Multiplication and Division -- These operations have the same precedence, so are performed in order of appearance, left to right. Of course, division is the same as multiplication by a reciprocal, and multiplication is a commutative and associative operation. Those features of these operations do not alter the "order of operations," but may alter your approach to actually doing an evaluation.
For example, 9*2/3 would be evaluated as (9*2)/3 = 18/3 = 6. However, recognizing that 9 = 3*3, you can rearrange the evaluation to ...
9/3*2 = 3*2 = 6
This rearrangement is allowed by the properties of multiplication, not by the Order of Operations.
You will also note that 9/3*2 is not the same as 9/(3*2). That is, the denominator in the division is only the first number after the division symbol. This is also true for expressions involving variables:
b/2a = (b/2)*a
If you want b/(2a), you must use parentheses.
Some authors make a distinction between the slash (/) and the symbol ÷ in their effect on an expression. The Order of Operations makes no such distinction, treating /, ÷, "over", "divided by" as all meaning exactly the same thing.
4. Addition and Subtraction -- These operations have the same precedence, so are performed in order of appearance, left to right. Of course, subtraction is the same as addition of an opposite, and addition is a commutative and associative operation. Those features of these operations do not alter the "order of operations," but may alter your approach to actually doing an evaluation.
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Based on the first letters of these operations, several mnemonic "words" or phrases have been invented to help you remember the order. Some are ...
PEMDAS
Please Excuse My Dear Aunt Sally
BIDMAS
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There are a number of tricky expressions floating around that test your understanding of the order of operations. Here is one of them:
10 × 4 - 2 × (4² ÷ 4) ÷ 2 ÷ 1/2 + 9
One of the things that makes this tricky is the distinction between ÷ and /, as discussed above. Here, the author of the expression intends for the / to indicate a fraction, so 2÷1/2 is intended to mean 2÷(1/2).
Working this according to the order of operations, we have ...
= 10 × 4 - 2 × (16 ÷ 4) ÷ 2 ÷ (1/2) + 9 . . . . . exponent inside parentheses
= 10 × 4 - 2 × 4 ÷ 2 ÷ (1/2) + 9 . . . . . division inside parentheses
= 40 - 2 × 4 ÷ 2 ÷ (1/2) + 9 . . . . . . first multiplication
= 40 - 8 ÷ 2 ÷ (1/2) + 9 . . . . . . second multiplication
= 40 - 4 ÷ (1/2) + 9 . . . . . first division
= 40 - 8 + 9 . . . . . . second division
= 32 . . . . . . first addition
= 41 . . . . . . second addition
