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Order of Operations
with Exponents

Master the rules for evaluating expressions that combine exponents, multiplication, division, addition, and subtraction: the key to getting every calculation right.

Pre-Algebra

What Are Exponents?

An exponent tells you how many times to multiply a number (the base) by itself. Instead of writing a long chain of multiplication, we use a compact notation.

\( 3 \)
\( 4 \)
 base exponent
\( = \)
\( 3 \times 3 \times 3 \times 3 \)
\( = 81 \)

Special Exponent Values

\( a^1 = a \)
Any number to the first power is just itself. Example: \( 7^1 = 7 \).
\( a^2 \) (Squared)
Multiply the base by itself twice. Example: \( 5^2 = 25 \).
\( a^3 \) (Cubed)
Multiply the base by itself three times. Example: \( 2^3 = 8 \).

Why Order Matters

Without a shared set of rules, the same expression can give different answers. Consider \( 2 + 3 \times 4^2 \):

✕ Left to right
\( 2 + 3 = 5 \), then \( 5 \times 4 = 20 \), then \( 20^2 = 400 \). Wrong.
✓ Correct order
\( 4^2 = 16 \), then \( 3 \times 16 = 48 \), then \( 2 + 48 = 50 \). Right!

The PEMDAS Rule

P E MD AS Parentheses Exponents Multiply / Divide (left to right) Add / Subtract (left to right) Highest priority Lowest priority Memory aid: "Please Excuse My Dear Aunt Sally"

Where Exponents Fit In

Exponents sit at the second-highest priority, right after parentheses and before multiplication or division. This means you must evaluate every exponent before you multiply, divide, add, or subtract.

1
Parentheses first: simplify anything inside \( ( \; ) \).
2
Exponents next: evaluate all powers like \( 3^2 \) or \( 5^3 \).
3
Multiply and Divide from left to right.
4
Add and Subtract from left to right.

Exponents vs. Multiplication

Students often confuse exponents with multiplication. They look similar but mean very different things:

Exponent Multiplication
Notation \( 3^4 \) \( 3 \times 4 \)
Meaning \( 3 \times 3 \times 3 \times 3 \) \( 3 + 3 + 3 + 3 \)
Result 81 12
Priority Higher (2nd) Lower (3rd)

How to Apply the Rules

1
Copy the expression. Rewrite it so you have room to work line by line.
2
Scan for parentheses. Simplify everything inside them first (applying PEMDAS inside, too).
3
Evaluate all exponents. Replace each power with its value.
4
Multiply and divide in order from left to right.
5
Add and subtract in order from left to right. You have your answer!

Example 1: Simple Expression

Evaluate: \( \; 5 + 2^3 \times 4 \)

E
Exponents first: \( 2^3 = 8 \). Expression becomes \( 5 + \textcolor{#fbbf24}{8} \times 4 \).
M
Multiply: \( 8 \times 4 = 32 \). Expression becomes \( 5 + \textcolor{#4ade80}{32} \).
A
Add: \( 5 + 32 = \textcolor{#f472b6}{37} \).

Example 2: Parentheses and Exponents

Evaluate: \( \; (3 + 1)^2 \times 5 - 10 \)

P
Parentheses: \( 3 + 1 = 4 \). Expression becomes \( \textcolor{#818cf8}{4}^2 \times 5 - 10 \).
E
Exponent: \( 4^2 = 16 \). Expression becomes \( \textcolor{#fbbf24}{16} \times 5 - 10 \).
M
Multiply: \( 16 \times 5 = 80 \). Expression becomes \( \textcolor{#4ade80}{80} - 10 \).
S
Subtract: \( 80 - 10 = \textcolor{#f472b6}{70} \).

Example 3: Multi-Step Challenge

Evaluate: \( \; 2^4 + 3 \times (10 - 6)^2 \div 8 \)

P
Parentheses: \( 10 - 6 = 4 \). Expression becomes \( 2^4 + 3 \times \textcolor{#818cf8}{4}^2 \div 8 \).
E
Exponents: \( 2^4 = 16 \) and \( 4^2 = 16 \). Expression becomes \( \textcolor{#fbbf24}{16} + 3 \times \textcolor{#fbbf24}{16} \div 8 \).
M/D
Multiply then divide (left to right): \( 3 \times 16 = 48 \), then \( 48 \div 8 = 6 \). Expression becomes \( 16 + \textcolor{#4ade80}{6} \).
A
Add: \( 16 + 6 = \textcolor{#f472b6}{22} \).

Visualizing the Evaluation Flow

Watch how \( 4 \times 2^3 + 6 \) simplifies step by step:

\( 4 \times 2^3 + 6 \)
Exponent
\( 4 \times \textcolor{#fbbf24}{8} + 6 \)
Multiply
\( \textcolor{#4ade80}{32} + 6 \)
Add
\( = 38 \)

Parentheses Change Everything

Placing parentheses in different spots can drastically change the answer. Compare these two expressions that use the same numbers:

\( 2 \times 3^2 \)
Exponent first: \( 3^2 = 9 \), then \( 2 \times 9 = \mathbf{18} \).
\( (2 \times 3)^2 \)
Parentheses first: \( 2 \times 3 = 6 \), then \( 6^2 = \mathbf{36} \).

Nested Parentheses with Exponents

Evaluate: \( \; (2 + (3^2 - 4)) \times 2 \)

1
Inner parentheses, exponent: \( 3^2 = 9 \). Inner becomes \( 9 - 4 = 5 \).
2
Outer parentheses: \( 2 + 5 = 7 \).
3
Multiply: \( 7 \times 2 = \textcolor{#4ade80}{14} \).

Common Mistakes to Avoid

✕ Multiplying the exponent
Writing \( 3^4 = 12 \) (base times exponent) instead of \( 3^4 = 81 \) (base times itself 4 times).
✕ Adding before exponents
In \( 5 + 2^3 \), doing \( 5 + 2 = 7 \) first. Exponents come before addition.
✕ Ignoring parentheses
Treating \( (2 + 3)^2 \) as \( 2 + 3^2 \). The parentheses force you to add first: \( 5^2 = 25 \), not \( 2 + 9 = 11 \).

Quick Reference Table

Step Operation Example Result
1 Parentheses \( (2+3) \) 5
2 Exponents \( 5^2 \) 25
3 Multiply / Divide \( 25 \times 2 \) 50
4 Add / Subtract \( 50 - 7 \) 43

Question 1 of 5

What is the value of \( 3 + 4^2 \)?

Question 2 of 5

Evaluate: \( (5 - 2)^3 \)

Question 3 of 5

What is the value of \( 2 \times 5^2 - 10 \)?

Question 4 of 5

Evaluate: \( 10 - 2^3 + 4 \times 3 \)

Question 5 of 5

What is the value of \( (1 + 2)^2 \times (8 - 6)^3 \)?