Exponents and Order of Operations

🔢 Exponents & Order of Operations

Part 1 of 5 — What Exponents Mean

Topics in This Part

🔑 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: $4 \times 4 \times 4 = 4^3$

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.

Examples

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

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 "$n$ cubed" — it gives the volume of a cube with edge $n$.

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

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.

Concept Check 🎯

Evaluating a Power Step by Step

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

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

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.

Evaluate the Powers 🧮

Write each power as a single number.

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

🔢 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

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:

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) $(-2)^5 = \,?$

🔢 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 $14$.

The Order of Operations

Do operations in this order:

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:

1. Pass 1: scan left to right and do every multiplication and division as you reach it.
2. 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) $7 \times 2 + 8 \div 4 = \,?$

🔢 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: $4 + 2 \times 3^2 = 4 + 2 \times 9 = 4 + 18 = 22$

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:

1. Is there anything inside parentheses or a fraction bar? Simplify it first.
2. Are there any exponents? Evaluate them next.
3. Multiply and divide, left to right.
4. 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) $\dfrac{3^2 + 7}{4} = \,?$

🔢 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

⚠️ 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

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.