Introduction to Exponents

An exponent is a shorthand way to write repeated multiplication. Instead of writing $2 \times 2 \times 2 \times 2$, you write $2^4$. This compact notation becomes essential as math gets more advanced — exponents appear in area and volume formulas, scientific notation, interest calculations, and electrical power formulas.

What Does an Exponent Mean?

An exponent tells you how many times to multiply the base by itself.

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

The two parts of exponential notation are:

• Base — the number being multiplied ($2$ in the example above)
• Exponent (also called the power) — how many times the base appears as a factor ($4$ in the example above)

We read $2^4$ as “two to the fourth power” or simply “two to the fourth.”

Expanded Form vs. Exponential Form

From expanded to exponential: Count how many times the base appears, then write it as the exponent.

From exponential to expanded: Write the base multiplied by itself the number of times shown by the exponent.

Example 1: Write in Exponential Form

Write $6 \times 6 \times 6$ in exponential form.

The base is 6 and it appears 3 times.

Answer: $6^3$

Example 2: Write in Expanded Form and Evaluate

Write $4^3$ in expanded form and find its value.

$4^3 = 4 \times 4 \times 4 = 64$

Answer: Expanded form is $4 \times 4 \times 4$; the value is $64$.

Special Powers: Exponent of 1

Any number raised to the first power equals itself.

$a^1 = a$

Examples: $5^1 = 5$, $100^1 = 100$, $(-3)^1 = -3$.

This makes sense: “multiply $a$ by itself once” just gives you $a$.

Special Powers: Exponent of 0

Any nonzero number raised to the zero power equals 1.

$a^0 = 1 \quad (\text{as long as } a \neq 0)$

Examples: $5^0 = 1$, $100^0 = 1$, $(-7)^0 = 1$.

Why does this work? Look at the pattern of dividing by the base each time:

$2^4 = 16$

$2^3 = 8 \quad (16 \div 2)$

$2^2 = 4 \quad (8 \div 2)$

$2^1 = 2 \quad (4 \div 2)$

$2^0 = 1 \quad (2 \div 2)$

Each time the exponent decreases by 1, you divide by the base. Following the pattern, $2^0 = 1$.

Powers of 10

Powers of 10 are especially important because our number system is base-10. Each power of 10 just adds a zero.

Quick rule: $10^n$ is a 1 followed by $n$ zeros.

Example 3: Powers of 10

What is $10^8$?

A 1 followed by 8 zeros: $100{,}000{,}000$ (one hundred million).

Answer: $10^8 = 100{,}000{,}000$

Evaluating Larger Powers

Example 4: Evaluate $3^5$

$3^5 = 3 \times 3 \times 3 \times 3 \times 3$

Work left to right:

$3 \times 3 = 9$

$9 \times 3 = 27$

$27 \times 3 = 81$

$81 \times 3 = 243$

Answer: $3^5 = 243$

Example 5: Evaluate $(-2)^4$

$(-2)^4 = (-2) \times (-2) \times (-2) \times (-2)$

$(-2) \times (-2) = 4$

$4 \times (-2) = -8$

$-8 \times (-2) = 16$

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

Key observation: A negative base raised to an even power gives a positive result. Raised to an odd power, the result is negative.

$(-2)^3 = (-2) \times (-2) \times (-2) = -8 \quad \text{(odd exponent, negative result)}$

$(-2)^4 = 16 \quad \text{(even exponent, positive result)}$

Watch Out: $(-2)^4$ vs. $-2^4$

These are not the same!

• $(-2)^4 = 16$ — the negative sign is inside the parentheses, so the entire base is $-2$.
• $-2^4 = -(2^4) = -16$ — the negative sign is outside, so you compute $2^4 = 16$ first, then apply the negative.

Real-World Application: Electrician — Wire in Conduit

Electricians sometimes calculate the cross-sectional area of a circular wire using $A = \pi r^2$, where $r$ is the radius. That $r^2$ means “radius squared” — the radius multiplied by itself.

If a wire has a radius of 0.1 inches:

$A = \pi \times (0.1)^2 = \pi \times 0.01 \approx 0.0314 \text{ square inches}$

Understanding exponents ensures electricians read and apply formulas correctly.

Real-World Application: Retail — Compound Growth

A store’s sales double every year. If Year 1 sales are $50,000, what are sales in Year 5?

$\text{Sales} = 50{,}000 \times 2^4 = 50{,}000 \times 16 = 800{,}000$

(The exponent is 4, not 5, because doubling happens 4 times to go from Year 1 to Year 5.)

Answer: Year 5 sales would be $800,000.

Common Mistakes to Avoid

1. Multiplying the base by the exponent. $3^4$ is not $3 \times 4 = 12$. It is $3 \times 3 \times 3 \times 3 = 81$.

2. Confusing $(-a)^n$ and $-a^n$. Parentheses matter. Without them, the negative sign is not part of the base.

3. Thinking $a^0 = 0$. Any nonzero number to the zero power is 1, not 0.

4. Forgetting negative base sign rules. Negative base with an even exponent gives a positive result; with an odd exponent, negative.

Practice Problems

Key Takeaways

• An exponent tells you how many times to multiply the base by itself: $a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}$.
• Any nonzero number to the zero power equals 1. Any number to the first power equals itself.
• $10^n$ is a 1 followed by $n$ zeros — a pattern that underpins our entire number system.
• Watch for the difference between $(-a)^n$ and $-a^n$ — parentheses change the meaning.
• Negative bases with even exponents give positive results; with odd exponents, negative results.

Return to Pre-Algebra for more topics in this section.