What an Exponent Is Telling You to Do
Whenever you spot an, it means "multiply a by itself, n times." The number getting multiplied - a - is the base. The small raised number, n, is the exponent, though you'll also hear people call it the power or the index.
3³ = 3 × 3 × 3 = 27
5² = 5 × 5 = 25 (say it as "5 squared")
2³ = 2 × 2 × 2 = 8 (say it as "2 cubed")
Only the second and third powers get special names - "squared" for 2 and "cubed" for 3. Every other power just uses "to the power of," so 4⁵ is read out loud as "4 to the power of 5," nothing fancier than that.
The Core Rules, One at a Time
1. Product Rule - Multiplying Same-Base Powers
When two powers share the same base and you're multiplying them, add the exponents.
2³ × 2⁴ = 2⁷ = 128
x² × x⁵ = x⁷
2. Quotient Rule - Dividing Same-Base Powers
Dividing works the mirror-image way: subtract the exponent in the denominator from the one in the numerator.
3⁶ ÷ 3² = 3⁴ = 81
x⁸ ÷ x³ = x⁵
3. Power Rule - A Power Raised to Another Power
If a power sits inside parentheses raised to a further power, multiply the two exponents together.
(2³)⁴ = 2¹² = 4096
(x²)⁵ = x¹⁰
4. Zero Exponent Rule
Any nonzero number raised to the power of 0 equals 1, full stop. It looks strange the first time around, but the quotient rule justifies it: aⁿ ÷ aⁿ works out to aⁿ⁻ⁿ, which is a⁰ - and any number divided by itself is 1.
7⁰ = 1
(xyz)⁰ = 1
0⁰ is left undefined (though a few contexts choose to define it as 1)
5. Negative Exponent Rule
A negative sign on an exponent isn't a sign flip on the answer - it's an instruction to take the reciprocal first, then raise that to the power shown.
2⁻³ = 1/2³ = 1/8 = 0.125
x⁻¹ = 1/x
3⁻² = 1/9
6. Fractional (Rational) Exponents
An exponent written as a fraction is really a root wearing a disguise. The denominator tells you which root to pull, and the numerator tells you what power to apply. Seeing this connection is useful because it shows roots and exponents are the same operation, just written in two different notations.
a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
8^(1/3) = ³√8 = 2
16^(3/4) = (⁴√16)³ = 2³ = 8
9^(1/2) = √9 = 3
7. Raising a Product or Quotient to a Power
(a/b)ⁿ = aⁿ / bⁿ
(2x)³ = 8x³
(3/4)² = 9/16
Cheat Sheet
| Rule | Formula | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | x³ × x² = x⁵ |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | x⁶ ÷ x² = x⁴ |
| Power of power | (aᵐ)ⁿ = aᵐⁿ | (x²)³ = x⁶ |
| Zero power | a⁰ = 1 | 5⁰ = 1 |
| Negative power | a⁻ⁿ = 1/aⁿ | 2⁻⁴ = 1/16 |
| Fractional power | a^(1/n) = ⁿ√a | 27^(1/3) = 3 |
Powers Worth Knowing From Memory
| Base | ² | ³ | ⁴ |
|---|---|---|---|
| 2 | 4 | 8 | 16 |
| 3 | 9 | 27 | 81 |
| 4 | 16 | 64 | 256 |
| 5 | 25 | 125 | 625 |
| 10 | 100 | 1000 | 10000 |
Real Places Exponents Show Up
Compound Interest
Compound interest grows the way it does because of an exponent - it's what lets each year's gain build directly on top of the last one instead of starting fresh:
A = final amount, P = principal, r = annual rate, n = years
£1000 at 5% for 10 years:
A = 1000 × (1.05)¹⁰ = 1000 × 1.629 = £1,629
Scientific Notation
Very large or very small numbers get rewritten as powers of 10 so nobody has to count zeros by eye. Light moves at 3 × 10⁸ meters per second (300,000,000 m/s), while a single hydrogen atom measures only about 1.2 × 10⁻¹⁰ meters across.
Computer Storage
Storage sizes are built on powers of 2 rather than powers of 10: a kilobyte is 2¹⁰, or 1024 bytes, and a gigabyte works out to 2³⁰, roughly 1.07 billion bytes. That mismatch is also why a drive labeled "1TB" reports a slightly smaller number once you actually plug it in - the manufacturer counted in powers of 10, but your operating system counts in powers of 2.
Mistakes That Show Up Again and Again
| Mistake | What People Write | Correct |
|---|---|---|
| Applying the product rule to different bases | 2³ × 3² = 6⁵ ✗ | 8 × 9 = 72 (there's no way to combine these) |
| Assuming a negative exponent gives a negative result | 2⁻³ = −8 ✗ | 2⁻³ = 1/8 |
| Adding exponents when the bases are being added | x² + x³ = x⁵ ✗ | These terms can't be combined - there's no rule for that |
| Mixing up the zero power | 3⁰ = 0 ✗ | 3⁰ = 1 |
Checking Your Work With a Calculator
Working through a big exponent by hand is a good way to catch mistakes, but you don't have to do it every time. Run the same problem through SolverCalc's exponent calculator - it's built to handle any base and power, negative and fractional exponents included, and it's essentially a dedicated calc solver for exactly this kind of question, so it's handy when the numbers get too big to track by hand.
Where That Leaves You
Every rule on this page traces back to one idea: an exponent is just repeated multiplication written in shorthand. Get comfortable with the product, quotient, and power rules first, since those three (add exponents, subtract exponents, multiply exponents) cover most of what you'll actually be asked to do. Zero, negative, and fractional exponents are the same logic seen from a different angle - they take a bit more thought the first few times, then settle in the same way the basic rules did. Because exponents keep reappearing in finance, science, and computing long after algebra class ends, this is one of the rare math topics where understanding the reasoning pays off more than memorizing the rules ever will.