Numerical Expressions
A numerical expression contains only numbers and operations — no variables, no equal sign. Something like . To evaluate it means to work it down to a single number, and everyone has to get the same number, which is why math has an agreed-on order of operations.
The order is short: grouping symbols first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Nearly every mistake on these problems comes from skipping a level or working right to left. Follow the order and the arithmetic takes care of itself.
The order of operations
1. Grouping symbols — parentheses, brackets, and fraction bars. Whatever sits inside a group gets evaluated first, as its own little problem.
2. Exponents. becomes before it touches anything else.
3. Multiplication and division, left to right. These two are the same rank — you do not finish all the multiplication before any division. You sweep across the expression once, handling whichever comes first.
4. Addition and subtraction, left to right. Same idea: equal rank, one sweep.
Fraction bars are grouping symbols
A wide fraction bar quietly groups everything above it and everything below it. In , you evaluate the whole top () and the whole bottom () separately, then divide. Treat the bar as if the top and the bottom each sat inside their own parentheses.
The left-to-right rule
The most common error is doing all the addition before any subtraction. In , working left to right gives . Adding first — , then — is wrong, because the subtraction came first in the line. Addition and subtraction are equal partners; take them in the order they appear.
Worked examples
Example 1: exponent, then divide, then left to right
Evaluate .
Answer:
Example 2: parentheses inside a bigger problem
Evaluate .
Answer:
Example 3: a fraction bar
Evaluate .
Answer:
Try one yourself
Common questions
Does multiplication always come before division?
No — they are the same rank. You work left to right and take whichever appears first. In , the division comes first, so the answer is , not .
What counts as a grouping symbol?
Parentheses , brackets , absolute value bars, and wide fraction bars. Anything inside one of these gets evaluated first, as its own separate problem.
What is the difference between a numerical expression and an algebraic expression?
A numerical expression has only numbers and operations, so it can be evaluated to a single number. An algebraic expression contains variables, like , so it can only be simplified — not evaluated — until you know what the variables equal.
Why did I get a different answer than my calculator?
Basic calculators evaluate as you type, left to right, ignoring the order of operations. Scientific calculators follow the real order. If you type into a basic calculator you get ; the correct value is .
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