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Order of Operations

Order of Operations for Algebraic Expressions

Before we can evaluate an expression we need to know the order in which the operations are done.

For all numerical or algebraic expressions, the order of evaluation is ( BEDMAS ):

.If an expression involves two or more operations at the same level of priority, those operations are done from left to right.

A few examples will show how these rules are applied.

y = 2x³

The operations to be done are multiply by 2 and cube. Exponents have 2nd prioity whereas multiply has 3rd priority. Given a value for x, we need to cube first then multiply by 2.

If we wish to multiply by 2 first then cube, we have to raise the priority of the multiplication by surrounding the multiplication by brackets, taking the priority to 1st. The expression for this different order is

y = (2x)³

Given the expression y = 7 + (4p/6 + 3)⁵ , we can see that the operations we are dealing with are add 7, brackets, multiply by 4, divide by 6 and raise to the power 5.  The stuff in the bracket must be done first.  Multiply and divide have the same 3rd priority so should be done in order from left to right, then the 4th priority add must be done.  Once the bracket is evaluated, we are left with raise to a power (2nd priority) and add (4th). Hence to evaluate y given a value for p we must multiply by 4 then divide by 6, add 3, raise to the power 5 and finally add 7.

Exercise

Given the expression    y = -4x² + 17, what is the order in which you would find y if you are given a value for x?

Two more things need consideration when considering order.

Nested Brackets

Nested brackets can be recognised by the appearance in the expression of 2 or more left brackets, ( , before any right brackets, ) .  The rule is do the innermost pair of brackets first and work your way outwards. For 3 sets of brackets:

(3rd priority(2nd priority(1st priority)more 2nd priority)more 3rd priority)

Example

z = 3(2a-4(bc)^-3)

The order in which we would evaluate z given the values of a, b and c is multiply b and c, raise to the power -3, multiply that by -4 then add that to 2 times a and finally multiply the lot by 3.

Implied Brackets

Many expressions in clsassical notation have no brackets when computer notation would demand we use brackets. 

Amongst these expressions are rational functions, .

The numerator and denominator are both expressions and their extent is shown by the length of the horizontal line.This implies that the numerator and denominator get evaluated first and finally the division is done. 

Strictly what we mean is . 

The brackets must be used in computer notationwith expressions:

.

Example

requires both (4x²) and (x + 1) to be evaluated before the division.  In computer notation this fraction is (4*x^2)/(x+1).

Another common implied bracket occurs when the root symbol is used.  Again although we do not usually put in the brackets, the length of the horizontal line tells the extent of the expression to take the root of.

Example

 is evaluated by raising 5 to the power of x, subtracting 22 and then taking the cube root.  The computer notation makes this explicit.

is equivalent to ((5^x)-22)^(1/3).

Exponents too have no brackets on functional powers.  The whole expression is simply superscripted to indicate it is the power.

Example

y^{2x + 1} is done in the order multiply x by 2 then add 1.  Finally raise y to this power.

y^{2x + 1} is equivalent to y^(2^x+1).

More Examples

Exercise

Show the operations required to evaluate the given expression starting from the variable. For each step, enter the operation in computer notation (e.g. /6) above the appropriate arrow. When finished, click "Next" to check your answer and continue. Type the operations:

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