Does PEMDAS mean multiplication always comes before division?

No. Multiplication and division are the same rank, and so are addition and subtraction. When operations of the same rank sit side by side, you perform them from left to right. So 18 3 · 2 = 12, not 3, and 10 - 4 + 1 = 7, not 5.

Why is -3^2 equal to -9 but (-3)^2 equal to 9?

An exponent attaches only to the symbol directly in front of it. In -3^2 that symbol is 3, so you square 3 to get 9 and then apply the negative, giving -9. In (-3)^2 the parentheses make -3 the entire base, so you multiply (-3)(-3) = 9. This distinction matters constantly when you substitute negative values into expressions such as -x^2.

How do I handle a fraction bar when there is no parenthesis anywhere?

Treat the fraction bar as if invisible parentheses surround the whole numerator and the whole denominator. Simplify the top to one number, simplify the bottom to one number, then divide. When you retype a fraction on a single line or into a calculator, add those parentheses yourself.

Should I simplify inside parentheses first, or distribute first?

If everything inside the parentheses is numbers, simplify inside first — it is faster and safer. If the inside contains variables that cannot be combined, such as 3(x + 5), distributing is the way forward. That case is the focus of the next lesson on the distributive property and like terms.

ALG1-1.2

Order of Operations & Evaluating Expressions

Master order of operations in Algebra 1: PEMDAS pitfalls, left-to-right rules, nested grouping, the fraction bar, and evaluating expressions at negative values.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Order of Operations & Evaluating Expressions, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

You already know that

2 + 3 \cdot 4

is 14, not 20. But algebra pushes that idea much further: expressions arrive with exponents on negative numbers, parentheses inside brackets, absolute value bars, and fraction bars that quietly group everything above and below them. If two people evaluate the same expression and get different numbers, one of them broke a rule of order — mathematics only works as a shared language because everyone agrees on the same reading order.

This lesson pins down that order exactly, including the two places where students most often slip: division and multiplication have equal rank (so do addition and subtraction), and

-x^2

does not mean the same thing as

(-x)^2

. You will then substitute given values — including negatives — into algebraic expressions and evaluate them cleanly. Everything here feeds directly into simplifying expressions, solving equations, and checking whether a proposed solution actually works.

The Order of Operations, Stated Precisely

Evaluating an expression means turning it into a single number. The agreed reading order is:

First, simplify inside grouping symbols, working from the innermost outward. Second, evaluate exponents (and roots). Third, do multiplication and division as they appear from left to right. Fourth, do addition and subtraction as they appear from left to right.

The memory device PEMDAS is useful but misleading if you read it as six separate steps. Multiplication does not outrank division, and addition does not outrank subtraction. Those pairs are ties, and ties are broken by reading left to right, exactly like reading a sentence.

Expression	Correct reading	Value	Common wrong reading	Wrong value
$12 \div 2 \cdot 3$	$(12 \div 2) \cdot 3$	$18$	$12 \div (2 \cdot 3)$	$2$
$20 - 8 + 5$	$(20 - 8) + 5$	$17$	$20 - (8 + 5)$	$7$
$36 \div 6 \div 3$	$(36 \div 6) \div 3$	$2$	$36 \div (6 \div 3)$	$18$

A cleaner way to think about the last stage: rewrite every subtraction as adding the opposite. Then

20 - 8 + 5

becomes

20 + (-8) + 5

, and because addition can be done in any order you cannot go wrong. The minus sign belongs to the number that follows it.

When you write your work, change only one thing per line and keep everything else copied exactly. Students who try to collapse three operations in a single step are the ones who lose track of a negative sign.

One more habit worth building now: never erase a grouping symbol until what is inside it is a single number, or until you have distributed across it deliberately.

Grouping Symbols: Nested, Invisible, and Disguised

Grouping symbols say "treat me as one quantity." Parentheses are the obvious ones, but four others behave identically and are easy to miss.

Symbol	Example	What it groups
Parentheses / brackets	$3[4 + 2(5 - 1)]$	Everything inside, innermost first
Fraction bar	$\dfrac{6 + 8}{9 - 2}$	The whole numerator and the whole denominator
Absolute value bars	$\lvert 3 - 11 \rvert$	The expression inside, before taking absolute value
Radical sign	$\sqrt{16 + 9}$	Everything under the bar
Exponent position	$2^{5 - 3}$	The entire exponent

With nested grouping, work from the inside out. In

3[4 + 2(5 - 1)]

: the innermost group gives

5 - 1 = 4

, then

2 \cdot 4 = 8

, then

4 + 8 = 12

, then

3 \cdot 12 = 36

.

The fraction bar is the sneakiest, because it groups without any visible parentheses.

\dfrac{6 + 8}{9 - 2}

means

(6 + 8) \div (9 - 2) = 14 \div 7 = 2

. If you type it into a calculator as 6 + 8 / 9 - 2, you get something entirely different, because the calculator sees only one division. Whenever you rewrite a fraction on one line, put parentheses around the top and around the bottom.

Absolute value bars group too: in

\lvert 3 - 11 \rvert

you must subtract first, getting

\lvert -8 \rvert = 8

. Taking absolute values of the pieces separately would give

\lvert 3 \rvert - \lvert 11 \rvert = 3 - 11 = -8

. Those are different results, so the grouping matters.

Substituting Values, Especially Negatives

To evaluate an algebraic expression at given values, replace each variable with its value and then follow the order of operations. The single most valuable habit: substitute inside parentheses every time.

If

x = -5

and the expression is

4x

, writing

4-5

is a disaster — it reads as subtraction and gives

-1

. Writing

4(-5)

gives the correct

-20

. Parentheses around every substituted value cost you two pen strokes and prevent the most common error in the unit.

The second trap is the difference between

-x^2

and

(-x)^2

. Exponents attach only to the symbol immediately beneath them. In

-x^2

, the exponent belongs to

x

alone, so you square first and then negate. In

(-x)^2

, the parentheses group the negative with the variable, so the whole quantity is squared.

Expression	Value at $x = 3$	Value at $x = -3$
$x^2$	$9$	$9$
$-x^2$	$-9$	$-9$
$(-x)^2$	$9$	$9$
$(2x)^2$	$36$	$36$
$2x^2$	$18$	$18$

Notice

-x^2

is negative even when

x

is negative: with

x = -3

you compute

(-3)^2 = 9

and then apply the leading minus, giving

-9

. Also compare

2x^2

with

(2x)^2

— in the first, only

x

is squared, so you square before multiplying by 2.

Finally, remember that a variable written next to a number or another variable means multiplication. In

5ab

with

a = -2

and

b = 4

, substitution gives

5(-2)(4) = -40

. Two negatives multiplied give a positive; three negatives multiplied give a negative — count the negative factors before you commit to a sign.

Building a Reliable Evaluation Routine

Speed comes from structure, not from skipping steps. A routine that works on nearly every problem in this unit:

Step one, rewrite the expression with parentheses around every value you substitute. Step two, evaluate every grouping symbol from the inside out, including numerator and denominator of any fraction. Step three, handle exponents, checking each time whether the base includes a negative sign. Step four, sweep left to right doing multiplications and divisions. Step five, sweep left to right doing additions and subtractions, or rewrite subtractions as adding the opposite.

Where students actually go wrong, in rough order of frequency: dropping a negative during substitution; treating

-x^2

(-x)^2

; doing addition before subtraction in a chain like

10 - 4 + 3

; ignoring the fraction bar's grouping power; and multiplying before clearing an inner set of parentheses.

A useful self-check is estimation. Before evaluating

\dfrac{2(11) - 1}{7}

, notice the numerator is around 21 and the denominator is 7, so the answer should be near 3. If your arithmetic produces

-14

, something structural went wrong, not just a digit.

Calculators follow these same rules, but they cannot see your intent. Typing 8/2(4) may or may not give what you expect depending on the model, and typing a fraction bar without parentheses almost never gives what you want. Type

(8 + 4)/(3 - 1)

with both sets of parentheses and the ambiguity disappears.

This routine reappears constantly: checking a solution to an equation means evaluating both sides at the candidate value, and evaluating a function at an input is exactly this same substitution process with new notation.

Key terms

Evaluate.: To replace each variable with a given value and simplify the result to a single number.
Order of operations.: The agreed sequence for simplifying: grouping symbols, then exponents, then multiplication and division left to right, then addition and subtraction left to right.
Grouping symbol.: Any notation that says "treat this as one quantity," including parentheses, brackets, the fraction bar, absolute value bars, and the radical sign.
Nested grouping.: Grouping symbols inside other grouping symbols, such as $2[3 + (7 - 4)]$ ; the innermost group is simplified first.
Fraction bar.: A division symbol that also groups: $\dfrac{a + b}{c - d}$ means $(a + b) \div (c - d)$ .
Base.: The quantity an exponent acts on. In $-x^2$ the base is $x$ ; in $(-x)^2$ the base is $-x$ .
Substitution.: Replacing a variable with a specified numerical value, best written with parentheses around that value.
Left-to-right rule.: The tie-breaker used when operations of equal rank appear in a row, as with $\div$ and $\cdot$ , or $+$ and $-$ .

Worked example

Evaluate

\dfrac{2a^2 - 3b}{\lvert a + b \rvert} + 4(b - a)^2

when

a = -3

and

b = 2

Start by substituting with parentheses around every value:

\frac{2(-3)^2 - 3(2)}{\lvert (-3) + (2) \rvert} + 4\big((2) - (-3)\big)^2

The fraction bar groups the numerator and the denominator, so handle each as its own mini-problem.

Numerator: exponents first,

(-3)^2 = 9

. Then multiply left to right:

2 \cdot 9 = 18

and

3 \cdot 2 = 6

. Then subtract:

18 - 6 = 12

.

Denominator: the absolute value bars group, so add inside first:

-3 + 2 = -1

. Then

\lvert -1 \rvert = 1

.

So the fraction is

12 \div 1 = 12

.

Second term: work inside the parentheses first.

2 - (-3) = 2 + 3 = 5

. Now the exponent:

5^2 = 25

. Now multiply:

4 \cdot 25 = 100

.

Last, add the two results:

12 + 100 = 112

.

Two checkpoints worth noticing. In the numerator,

2a^2

means square

a

first and then double it — squaring

2a

instead would have given

(2a)^2 = (-6)^2 = 36

and a numerator of

36 - 6 = 30

. And in

2 - (-3)

, subtracting a negative adds; students who write

2 - 3 = -1

end up with

4(-1)^2 = 4

and a final answer of 16.

Practice questions

Evaluate

-x^2 + 3(x - 5)

when

x = -4

$-43$
$-11$
$11$
$43$

Answer: $-43$

Substitute with parentheses:

-(-4)^2 + 3\big((-4) - 5\big)

. The exponent applies only to

-4

inside its own parentheses, giving

16

, and the leading minus sign then makes that term

-16

. Inside the second grouping,

-4 - 5 = -9

, and

3(-9) = -27

. Finally

-16 + (-27) = -43

. The value

-11

comes from mistakenly reading

-x^2

(-x)^2 = 16

, which would give

16 - 27

. Remember that in

-x^2

the base of the exponent is just

x

Evaluate

\dfrac{4 + 2 \cdot 6}{2^3 - 4}

$1.5$
$4$
$12$
$16$

Answer: $4$

The fraction bar groups the entire top and the entire bottom. Numerator: multiply before adding,

2 \cdot 6 = 12

, then

4 + 12 = 16

. Denominator:

2^3 = 8

, then

8 - 4 = 4

. So the value is

16 \div 4 = 4

. The answer

1.5

comes from rewriting the expression on one line as

4 + 2 \cdot 6 \div 2^3 - 4

without parentheses, which changes what the division acts on.

Maya says that

24 \div 4 \cdot 3 = 2

because "multiplication comes before division in PEMDAS." Explain what is wrong with her reasoning, give the correct value, and describe one habit that prevents this error.

Answer: Maya is wrong: multiplication and division have equal priority, so they are performed left to right. Reading left to right, $24 \div 4 = 6$ and then $6 \cdot 3 = 18$ . Her answer of 2 comes from computing $4 \cdot 3 = 12$ first and then $24 \div 12$ .

PEMDAS lists M before D only because the acronym needs some order to be pronounceable; the two operations are a single tied rank, exactly like addition and subtraction. A helpful habit is to rewrite the expression using multiplication by a reciprocal, so

24 \div 4 \cdot 3

becomes

24 \cdot \frac{1}{4} \cdot 3

. Written that way there is no ambiguity, and multiplying in any order gives 18. A second habit is to underline each tied pair as you sweep left to right so you never jump ahead to an operation further right.

FAQ

Does PEMDAS mean multiplication always comes before division?: No. Multiplication and division are the same rank, and so are addition and subtraction. When operations of the same rank sit side by side, you perform them from left to right. So $18 \div 3 \cdot 2 = 12$ , not 3, and $10 - 4 + 1 = 7$ , not 5.
Why is $-3^2$ equal to $-9$ but $(-3)^2$ equal to $9$ ?: An exponent attaches only to the symbol directly in front of it. In $-3^2$ that symbol is 3, so you square 3 to get 9 and then apply the negative, giving $-9$ . In $(-3)^2$ the parentheses make $-3$ the entire base, so you multiply $(-3)(-3) = 9$ . This distinction matters constantly when you substitute negative values into expressions such as $-x^2$ .
How do I handle a fraction bar when there is no parenthesis anywhere?: Treat the fraction bar as if invisible parentheses surround the whole numerator and the whole denominator. Simplify the top to one number, simplify the bottom to one number, then divide. When you retype a fraction on a single line or into a calculator, add those parentheses yourself.
Should I simplify inside parentheses first, or distribute first?: If everything inside the parentheses is numbers, simplify inside first — it is faster and safer. If the inside contains variables that cannot be combined, such as $3(x + 5)$ , distributing is the way forward. That case is the focus of the next lesson on the distributive property and like terms.

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