Integer Exponents — I can use integer exponents to represent repeated multiplication and evaluate expressions. - Pre Algebra

SECTION 1

Where Did Exponents Come From?

Imagine writing 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 every time you needed to use that number. That's ten 2s multiplied together! People in ancient times ran into this exact problem. As math grew more complex, mathematicians needed a shortcut. That shortcut is what we now call exponents (a small number written above and to the right of another number to show how many times to multiply it by itself).

~1600 BCE • Ancient Babylonians

Babylonian scribes used tables of squares and cubes on clay tablets to help with calculations. They didn't have exponent notation, but they understood repeated multiplication.

~250 CE • Diophantus of Alexandria

The Greek mathematician Diophantus used special symbols for squares and cubes in his book Arithmetica. This was one of the first times repeated multiplication got its own notation.

1637 • René Descartes

The French mathematician Descartes introduced the modern superscript notation we use today, such as x² and x³. This is the same style you'll learn in this lesson!

1748 • Leonhard Euler

Euler extended exponents to include negative numbers and zero. He showed that the rules of exponents work for all integers, not just positive ones.

So here's the big question this lesson answers: How can we write repeated multiplication in a short, clear way—and what happens when the exponent is zero or negative?

SECTION 2

Core Principles of Exponents

Before you start calculating, you need to know the vocabulary. An expression like 5³ has two parts. The big number on the bottom is the base (the number being multiplied). The small number on the top right is the exponent (how many times the base is used as a factor). The whole thing is called a power. So 5³ is read as "five to the third power."

Positive Exponents

A positive exponent tells you how many times to multiply the base by itself. For example, 3⁴ means 3 × 3 × 3 × 3 = 81.

Zero Exponent

Any nonzero number raised to the zero power equals 1. For example, 7⁰ = 1. This might seem strange, but it follows a logical pattern you'll see soon!

Negative Exponents

A negative exponent means "take the reciprocal." For example, 2⁻³ = 1 / 2³ = 1/8. The negative sign flips the base to the bottom of a fraction.

Special Exponents: 1 and 2

Any number to the first power is just itself (9¹ = 9). A number to the second power is called "squared" (4² = 16), and to the third power is called "cubed" (4³ = 64).

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KEY TAKEAWAY

Think of exponents like a recipe shortcut. If a cookie recipe says "add 1 cup of sugar" five times, you could just write "add 5 cups of sugar." Exponents do the same thing for multiplication: instead of writing 2 × 2 × 2 × 2 × 2, you write 2⁵. Same answer, way less writing!

SECTION 3

Seeing Exponents in Action

The diagram below shows how the expression 2⁵ breaks down. Notice how the base (2) is written as a factor five times, matching the exponent. Each multiplication step doubles the result.

Each colored box represents one factor of 2. The tree below shows how the running product doubles at every step, ending at 32.

Notice the pattern: each time you multiply by the base again, the result grows quickly. That's why exponents are sometimes called "power notation"—they pack a lot of power into a tiny symbol!

SECTION 4

The Rules of Integer Exponents

Now let's look at the key formulas. These rules work for any base (as long as the base isn't zero when the exponent is zero or negative).

POSITIVE EXPONENT

aⁿ = a × a × a × … × a (n factors)

Here, a is the base and n is the exponent (a positive whole number). You multiply the base by itself n times.

ZERO EXPONENT

a⁰ = 1 (where a ≠ 0)

Any nonzero number raised to the zero power equals 1. Think of it this way: 2³ = 8, 2² = 4, 2¹ = 2. Each time you subtract 1 from the exponent, you divide by 2. So 2⁰ = 2 ÷ 2 = 1.

NEGATIVE EXPONENT

a⁻ⁿ = 1 / aⁿ (where a ≠ 0)

A negative exponent flips the base into the denominator of a fraction. For example, 5⁻² = 1 / 5² = 1 / 25.

💡

Why does a⁰ = 1?

Look at the pattern: 3⁴ = 81, 3³ = 27, 3² = 9, 3¹ = 3. Each time the exponent drops by 1, you divide by 3. Following that pattern, 3⁰ = 3 ÷ 3 = 1. The pattern keeps working for negative exponents too: 3⁻¹ = 1 ÷ 3 = 1/3.

SECTION 5

The Powers-of-Two Pattern Table

One of the best ways to understand exponents is to see the pattern from negative exponents through zero and up to positive exponents. The table below uses base 2 to show how values change.

Exponent Form	Expanded Form	Value
2⁻³	1 / (2 × 2 × 2)	1/8 = 0.125
2⁻²	1 / (2 × 2)	1/4 = 0.25
2⁻¹	1 / 2	1/2 = 0.5
2⁰	(any nonzero number)⁰	1
2¹	2	2
2²	2 × 2	4
2³	2 × 2 × 2	8
2⁴	2 × 2 × 2 × 2	16
2⁵	2 × 2 × 2 × 2 × 2	32

Powers of 2 from exponent −3 to 5

This number line shows how powers of 2 behave. Negative exponents (pink) produce fractions. The zero exponent (gold) equals 1. Positive exponents (green and cyan) grow rapidly.

Notice the dividing-by-2 pattern as you move left: 32 → 16 → 8 → 4 → 2 → 1 → 1/2 → 1/4 → 1/8. Each step left divides by the base. Each step right multiplies by the base. This pattern works for every base, not just 2!

SECTION 6

Worked Example: Evaluating Expressions with Exponents

Let's work through a problem step by step. We'll evaluate the expression (−3)⁴ and then tackle 4⁻² so you can see both positive and negative exponents in action.

Example 1: Evaluate (−3)⁴

1
Step 1 — Identify the base and exponent

The base is −3 (including the negative sign, because it's inside the parentheses). The exponent is 4. This means we multiply −3 by itself 4 times.
2
Step 2 — Write out the expanded form

(−3)⁴ = (−3) × (−3) × (−3) × (−3)
3
Step 3 — Multiply step by step

First pair: (−3) × (−3) = 9. A negative times a negative is positive. Second pair: (−3) × (−3) = 9. Now multiply the results: 9 × 9 = 81.

(−3)⁴ = 81
4
Step 4 — Check the sign

The exponent is even (4), so the answer is positive. Remember: an even number of negative factors always gives a positive result.

Example 2: Evaluate 4⁻²

1
Step 1 — Recognize the negative exponent

The base is 4 and the exponent is −2. A negative exponent means we take the reciprocal (flip it into a fraction).
2
Step 2 — Rewrite using the rule a⁻ⁿ = 1 / aⁿ

4⁻² = 1 / 4²
3
Step 3 — Evaluate the positive exponent

4² = 4 × 4 = 16. So 4⁻² = 1 / 16.

4⁻² = 1/16

SECTION 7

Common Mistakes & How to Avoid Them

Exponents look simple, but there are a few traps that catch students all the time. The table below shows the most common mistakes and the correct way to think about each one.

Common Mistake	Why It's Wrong	Correct Answer
−3² = 9	Without parentheses, only the 3 is squared. The negative sign stays in front.	−3² = −(3²) = −9
2⁰ = 0	The zero exponent rule says any nonzero base to the zero power is 1, not 0.	2⁰ = 1
5⁻² = −25	A negative exponent doesn't make the answer negative. It means reciprocal (1 divided by the power).	5⁻² = 1/25
3⁴ = 12	This is 3 × 4 (multiplication), not 3⁴ (exponentiation). Exponents mean repeated multiplication, not just one multiplication.	3⁴ = 3 × 3 × 3 × 3 = 81

Watch out for these four common exponent errors!

⚠️

KEY TAKEAWAY

Parentheses matter a lot with exponents! Think of it like a sports jersey: (−3)² means the whole player (negative sign and all) is being squared. But −3² means only the number 3 is wearing the "squared" jersey, and the negative sign is just standing on the sideline.

SECTION 8

From Integer Exponents to More Advanced Ideas

Integer exponents are your first step into a bigger world. As you move through algebra and beyond, you'll see exponents everywhere. Here's a preview of where they lead.

What You Know Now	What's Coming Next
Positive integer exponents (like 5³)	Exponent rules for multiplying and dividing powers (like 2³ × 2⁴ = 2⁷)
Zero exponent (a⁰ = 1)	Scientific notation, where 10⁰ helps you express very large and very small numbers
Negative exponents (a⁻ⁿ = 1/aⁿ)	Fractional exponents (like $8^{1/3}$ = ∛8 = 2), which connect exponents to roots
Evaluating expressions like (−2)⁴	Exponential functions and graphs, which model population growth, compound interest, and radioactive decay

Your exponent journey is just getting started!

The good news? Every one of these advanced topics builds directly on the rules you're learning right now. Master integer exponents, and you'll have a solid foundation for all of it.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

In the expression 7³, identify the base, the exponent, and explain in your own words what the expression means.

Show answer

The base is 7 and the exponent is 3. The expression 7³ means "multiply 7 by itself 3 times," which is 7 × 7 × 7. The exponent tells you how many times the base appears as a factor. So 7³ = 343. Remember, an exponent is not the same as multiplying by the exponent—7³ is not 7 × 3 = 21.

Question 2: BASIC CALCULATION

Evaluate 10⁴.

Show answer

Using the definition of a positive exponent, write out 10⁴ = 10 × 10 × 10 × 10. Multiply step by step: 10 × 10 = 100, then 100 × 10 = 1,000, then 1,000 × 10 = 10,000. So 10⁴ = 10,000. Notice that the exponent on a base of 10 tells you how many zeros to write after the 1—this is why powers of 10 are so useful in science!

Question 3: INTERMEDIATE

Evaluate 6⁻² and express your answer as both a fraction and a decimal.

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A negative exponent means reciprocal, so 6⁻² = 1 / 6². First evaluate 6² = 6 × 6 = 36. Then 6⁻² = 1/36. To convert to a decimal, divide 1 ÷ 36 ≈ 0.0278 (rounded). The negative exponent did not make the answer negative—it moved the base to the denominator of a fraction.

Question 4: APPLIED

A colony of bacteria doubles every hour. If you start with 1 bacterium, you can find the number of bacteria after h hours using the expression 2ʰ. How many bacteria are there after 8 hours?

Show answer

Substitute h = 8 into the expression: 2⁸. Now evaluate: 2⁸ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. You can build up: 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64, 2⁷ = 128, 2⁸ = 256. After 8 hours there are 256 bacteria. This is an example of exponential growth, where exponents model how quantities can increase very quickly.

Question 5: CRITICAL THINKING

Without calculating each one fully, determine which is greater: (−2)⁵ or (−2)⁶. Explain your reasoning using what you know about even and odd exponents.

Show answer

(−2)⁶ is greater. Here's why: when a negative base has an odd exponent (like 5), the result is negative because there's an unpaired negative factor. So (−2)⁵ = −32 (a negative number). When a negative base has an even exponent (like 6), the negatives pair up and cancel, giving a positive result. So (−2)⁶ = 64 (a positive number). Any positive number is greater than any negative number, so (−2)⁶ > (−2)⁵. This even/odd pattern is a powerful shortcut for working with negative bases.

SUMMARY

Lesson Summary

Summary

An exponent is a shorthand for repeated multiplication. In a power like aⁿ, the base (a) is the number being multiplied, and the exponent (n) tells you how many times. A positive exponent means multiply the base by itself that many times (3⁴ = 81). The zero exponent rule says any nonzero number raised to the zero power equals 1 (a⁰ = 1). A negative exponent means reciprocal: a⁻ⁿ = 1 / aⁿ.

Watch out for parentheses with negative bases—(−3)² = 9 but −3² = −9. Also remember that a negative base raised to an even exponent gives a positive result, while an odd exponent gives a negative result. Integer exponents are the foundation for scientific notation, exponent rules, and exponential functions you'll meet in algebra and beyond.