🎯⭐ INTERACTIVE LESSON

Exponents and Order of Operations

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Exponents and Order of Operations - Complete Interactive Lesson

Part 1: What Exponents Mean

🔢 Exponents & Order of Operations

Part 1 of 5 — What Exponents Mean

Topics in This Part

Section
Repeated Multiplication
Base and Exponent
Reading "Squared" and "Cubed"

🔑 Key Concept: An exponent is just a shortcut for repeated multiplication. Instead of writing $2 \times 2 \times 2 \times 2 \times 2$ , we write $2^5$ . Same number — fewer pencil strokes.

Repeated Multiplication

Multiplication is a shortcut for repeated addition: $4 + 4 + 4 = 3 \times 4$

An exponent is a shortcut for repeated multiplication: $4 \times 4 \times 4 = 4^3$

Name the Parts 🔽

In the power $7^5$ , identify each piece.

"Squared" and "Cubed"

Two exponents are so common they get their own names:

$n^2$ is read " $n$ squared" — it gives the area of a square with side $n$ .
$n^3$ is read — it gives the volume of a cube with edge .

Concept Check 🎯

Evaluating a Power Step by Step

To turn a power into a plain number, multiply two factors at a time:

$2^4 = \underbrace{2 \times 2}_{4} \times 2 \times 2 = 4 \times 2 \times 2 = 8 \times 2 = 16$

Evaluate the Powers 🧮

Write each power as a single number.

1) $2^4 = \,?$ 2) $5^3 = \,?$ 3) $10^{4}$

Part 2: Special Powers & Negative Bases

🔢 Exponents & Order of Operations

Part 2 of 5 — Special Powers & Negative Bases

🔑 The Idea: A few exponent rules look strange the first time — powers of $1$ , exponents of $0$ , and negative bases. Once you see why each works, they stop being tricky.

Three Special Cases

Rule	Why	Example
Any base to the first power is itself	one factor	$9^1 = 9$

Part 3: The Order of Operations (PEMDAS)

🔢 Exponents & Order of Operations

Part 3 of 5 — The Order of Operations (PEMDAS)

🔑 Why we need it: $2 + 3 \times 4$ could be $20$ (if you add first) or $14$ (if you multiply first). A single agreed-upon order guarantees everyone gets the same answer. The right answer here is $14$ .

Part 4: Putting Exponents Into the Order

🔢 Exponents & Order of Operations

Part 4 of 5 — Putting Exponents Into the Order

🔑 The Big Combination: Now we fold exponents (the E in PEMDAS) and grouping symbols together. Exponents come right after parentheses — before any multiply, divide, add, or subtract.

Exponents Come Early

Example A: $4 + 2 \times 3^2$

Exponent first, then multiply, then add: $4 + 2 \times 3^{2} = 4 + 2 \times 9 = 4 + 18$

Part 5: Mixed Practice & Mastery Check

🔢 Exponents & Order of Operations

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) read and evaluate exponents, (2) handle negative bases and parentheses, (3) apply PEMDAS, and (4) combine them in one expression. Let's put it all together.

Quick Reference

Situation	Key move
$a^n$	multiply $a$ by itself $n$ times
(with )

In the symbol $4^3$ :

the big number $4$ is the base — the number being multiplied,
the small raised number $3$ is the exponent (or power) — how many times you write the base.

Power	Meaning	Value
$2^3$	$2 \times 2 \times 2$	$8$
$5^2$	$5 \times 5$	$25$
$3^4$	$3 \times 3 \times 3 \times 3$	$81$
$10^3$	$10 \times 10 \times 10$	$1000$

⚠️ Watch out: $4^3$ does not mean $4 \times 3 = 12$ . The exponent counts how many $4$ 's you multiply, so $4^3 = 4 \times 4 \times 4 = 64$ .

$5^2 = 25 \quad (\text{"five squared"}) \qquad 2^3 = 8 \quad (\text{"two cubed"})$

For any other exponent we say "to the power of": $4^5$ is "four to the fifth power."

💡 Perfect squares are values of $n^2$ : $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$ . Memorizing these speeds up almost every future topic.

For powers of ten there's a shortcut: the exponent equals the number of zeros.

$10^2 = 100, \quad 10^3 = 1000, \quad 10^5 = 100000$

Now try evaluating a few yourself.

A quick way to feel the $n^0 = 1$ rule: look at a shrinking pattern of powers of $2$ . Each step down divides by $2$ :

$2^3 = 8,\quad 2^2 = 4,\quad 2^1 = 2,\quad 2^0 = 1$

Going $8 \to 4 \to 2$ , the next step is $2 \div 2 = 1$ , so $2^0 = 1$ .

⚠️ The only exception people argue about is $0^0$ ; in Pre-Algebra we won't use it. But $0^1 = 0$ , $0^2 = 0$ , and $5^0 = 1$ are all standard.

Concept Check 🎯

Negative Bases & Parentheses

The grouping symbols around a negative number decide whether the minus sign is part of the base.

Case 1: the negative is inside parentheses

$(-2)^4 = (-2)(-2)(-2)(-2) = 16$

Here the base is $-2$ , multiplied four times. A negative base raised to an even power is positive; raised to an odd power it is negative:

$(-2)^3 = (-2)(-2)(-2) = -8$

Case 2: no parentheses

$-2^4 = -(2^4) = -(16) = -16$

Without parentheses, the exponent applies only to the $2$ , and the minus sign sits out front like a "negative of."

⚠️ This is the single most common exponent mistake. $(-2)^4 = 16$ but $-2^4 = -16$ . The parentheses change the answer.

Sign Check 🔽

Decide the sign and value of each power.

A One-Glance Sign Rule

Before you compute, you can already predict the sign of a power:

Base	Exponent	Sign of result
positive	anything	positive
negative (in parentheses)	even	positive
negative (in parentheses)	odd	negative

So $(-5)^2$ will be positive and $(-5)^3$ will be negative — before you multiply anything. Use this to check yourself on the drill below.

Evaluate Carefully 🧮

Watch the parentheses on each one.

1) $(-4)^2 = \,?$ 2) $-4^2 = \,?$ 3) $(-2)^5 = \,?$

The Order of Operations

Do operations in this order:

Step	Operation	Memory word
1	Parentheses (grouping symbols)	Please
2	Exponents	Excuse
3	Multiplication & Division (left → right)	My Dear
4	Addition & Subtraction (left → right)	Aunt Sally

This is often abbreviated PEMDAS ("Please Excuse My Dear Aunt Sally").

⚠️ Two pairs are tied. Multiplication and division have the same priority — do them left to right as they appear. Same for addition and subtraction. PEMDAS does not mean "all multiplication before all division."

Concept Check 🎯

Worked Examples

Example A: $2 + 3 \times 4$

Multiplication comes before addition: $2 + 3 \times 4 = 2 + 12 = 14$

Example B: $20 - 6 \div 2$

Division comes before subtraction: $20 - 6 \div 2 = 20 - 3 = 17$

Example C: $12 \div 3 \times 2$

Multiplication and division tie, so go left to right: $12 \div 3 \times 2 = 4 \times 2 = 8$

⚠️ If you did the multiplication first here, you'd get $12 \div 6 = 2$ — wrong. Left-to-right matters when operations are tied.

What Goes First? 🔽

For each expression, pick the operation you must do first.

A Two-Pass Strategy

A reliable way to evaluate a longer expression:

Pass 1: scan left to right and do every multiplication and division as you reach it.
Pass 2: scan left to right again and do every addition and subtraction.

$7 \times 2 + 8 \div 4 \;\xrightarrow{\text{pass 1}}\; 14 + 2 \;\xrightarrow{\text{pass 2}}\; 16$

This keeps the tied operations in the correct left-to-right order. Try it on the drill below.

Evaluate (no exponents yet) 🧮

1) $9 + 4 \times 3 = \,?$ 2) $30 \div 5 - 2 = \,?$ 3) $7 \times 2 + 8 \div 4 = \,?$

4 + 2 \times 3^{2} = 4 + 2 \times 9 = 4 + 18 = 22

Example B: $5 \times 2^3 - 6$

$5 \times 2^3 - 6 = 5 \times 8 - 6 = 40 - 6 = 34$

Example C: $(4 + 1)^2$ vs. $4 + 1^2$

Grouping changes everything: $(4 + 1)^2 = 5^2 = 25 \qquad 4 + 1^2 = 4 + 1 = 5$

💡 The order inside PEMDAS: Parentheses → Exponents → Multiply/Divide → Add/Subtract. The exponent always waits for what's inside its parentheses to finish, but runs before the multiplication and addition around it.

Concept Check 🎯

Grouping Symbols & Fraction Bars

Parentheses $()$ , brackets $[\,]$ , and the fraction bar all act as grouping symbols. Work from the innermost group outward.

Example: $3 \times [\,(8 - 5)^2 + 1\,]$

$3 \times [\,(8-5)^2 + 1\,] = 3 \times [\,3^2 + 1\,] = 3 \times [\,9 + 1\,] = 3 \times 10 = 30$

Example: a fraction bar groups top and bottom

$\frac{6 + 2^3}{7} = \frac{6 + 8}{7} = \frac{14}{7} = 2$

The bar tells you to finish the whole numerator ( $6 + 8 = 14$ ) before dividing by $7$ .

⚠️ Treat a fraction bar like hidden parentheses around the top and around the bottom: simplify each completely, then divide.

Step Order 🔽

You're evaluating $1 + 2 \times 3^2$ . Choose the result after each stage.

Your PEMDAS Checklist

For any expression that mixes grouping, exponents, and operations, ask in order:

Is there anything inside parentheses or a fraction bar? Simplify it first.
Are there any exponents? Evaluate them next.
Multiply and divide, left to right.
Add and subtract, left to right.

💡 If you ever feel stuck, rewrite one step per line — finishing exactly one operation each line makes mistakes easy to spot.

Full PEMDAS Drill 🧮

1) $5^2 - 4 \times 3 = \,?$ 2) $2 \times (1 + 4)^2 = \,?$ 3) $\dfrac{3^2 + 7}{4} = \,?$

⚠️ Top traps: $4^3 \ne 12$ (it's $64$ ); $-2^4 \ne (-2)^4$ ; and M/D and A/S are read left to right, not "all M before all D."

Mixed Practice 🎯

Before the Final Drill — Avoid These Traps

Trap	Wrong	Right
Reading $a^b$ as $a \times b$	$4^3 = 12$	$4^3 = 64$
Squaring a sum term-by-term	$(2+3)^2 = 13$	$(2+3)^2 = 25$
Forgetting the fraction bar groups	$\dfrac{6+8}{7} = 6 + \dfrac{8}{7}$
Ignoring left-to-right ties	$12 \div 3 \times 2 = 2$	$= 8$

Keep these in mind as you finish the drill below.

Final Drill 🧮

1) $4 \times 3^2 - 10 = \,?$ 2) $\dfrac{2^4 + 4}{5} = \,?$ 3) $2 \times (6 - 2)^2 + 7 = \,?$

You're Ready

You can now read a power, evaluate it (signs and all), and march through any expression with PEMDAS. One last check to lock it in.

🔑 Remember the order: Parentheses → Exponents → Multiply/Divide (left→right) → Add/Subtract (left→right).

Exit Quiz ✅

Answer all three to finish the lesson.

Exponents and Order of Operations

Exponents and Order of Operations - Complete Interactive Lesson

Part 1: What Exponents Mean

🔢 Exponents & Order of Operations

Topics in This Part

Repeated Multiplication

"Squared" and "Cubed"

Evaluating a Power Step by Step

Part 2: Special Powers & Negative Bases

🔢 Exponents & Order of Operations

Three Special Cases

Part 3: The Order of Operations (PEMDAS)

🔢 Exponents & Order of Operations

Part 4: Putting Exponents Into the Order

🔢 Exponents & Order of Operations

Exponents Come Early

Example A: 4+2×324 + 2 \times 3^24+2×32

Part 5: Mixed Practice & Mastery Check

🔢 Exponents & Order of Operations

Quick Reference

Examples

Negative Bases & Parentheses

Case 1: the negative is inside parentheses

Case 2: no parentheses

A One-Glance Sign Rule

The Order of Operations

Worked Examples

Example A: 2+3×42 + 3 \times 42+3×4

Example B: 20−6÷220 - 6 \div 220−6÷2

Example C: 12÷3×212 \div 3 \times 212÷3×2

A Two-Pass Strategy

Example B: 5×23−65 \times 2^3 - 65×23−6

Example C: (4+1)2(4 + 1)^2(4+1)2 vs. 4+124 + 1^24+12

Grouping Symbols & Fraction Bars

Example: 3×[ (8−5)2+1 ]3 \times [\,(8 - 5)^2 + 1\,]3×[(8−5)2+1]

Example: a fraction bar groups top and bottom

Your PEMDAS Checklist

Before the Final Drill — Avoid These Traps

You're Ready

Example A: $4 + 2 \times 3^2$

Example A: $2 + 3 \times 4$

Example B: $20 - 6 \div 2$

Example C: $12 \div 3 \times 2$

Example B: $5 \times 2^3 - 6$

Example C: $(4 + 1)^2$ vs. $4 + 1^2$

Example: $3 \times [\,(8 - 5)^2 + 1\,]$