Calculate powers and exponents instantly. Solve positive, negative, and fractional exponents with step-by-step solutions. Free online exponent calculator with all exponent rules, formulas, and real-world examples.
Enter values into any two of the input fields to solve for the third.
Exponentiation is a mathematical operation, written as an, involving the base a and an exponent n. Think of an exponent as a "repeat counter" — it tells you how many times to use the base number in a multiplication. In the case where n is a positive integer, exponentiation corresponds to repeated multiplication of the base, n times.
So, 53 means 5 × 5 × 5. That's 125. Easy, right?
The base can be any number – whole numbers, fractions, decimals, even negative numbers. The exponent can be any number too. But when the exponent isn't a simple positive whole number, things get a little weird. We'll get to that.
The calculator above accepts negative bases, but does not compute imaginary numbers. It also does not accept fractions, but can be used to compute fractional exponents, as long as the exponents are input in their decimal form.
Exponents have their own set of rules. Once you know these, you can solve almost any exponent problem. Let's break them down.
When you multiply powers with the same base, just add the exponents. If you have 2³ × 2⁴, you're really just doing (2×2×2) × (2×2×2×2). That's 2 multiplied by itself 7 times. So 3 + 4 = 7. Answer: 2⁷ = 128.
When you divide powers with the same base, subtract the exponents. If you have 5⁶ ÷ 5², you're canceling out two of the 5's. So 6 - 2 = 4. Answer: 5⁴ = 625.
(3²)³ means take 3² (which is 9) and raise it to the 3rd power. So 9³ = 729. The shortcut? Just multiply the exponents: 2 × 3 = 6. So 3⁶ = 729. Same answer, less work.
Anything (except 0) to the power of 0 is 1. Why? Think about the quotient rule: 2³ ÷ 2³ = 2⁽³⁻³⁾ = 2⁰. But 2³ ÷ 2³ = 8 ÷ 8 = 1. So 2⁰ must equal 1. It's not magic — it's just math being consistent.
A negative exponent doesn't make the answer negative. It means "one divided by" the positive exponent. So 2⁻³ = 1 / 2³ = 1/8 = 0.125. See? Not negative at all. It's just a fraction.
(2 × 4)² = 8² = 64 | (2 × 4)² = 2² × 4² = 4 × 16 = 64
(2/5)² = 2²/5² = 4/25
Any number to the power of 1 is itself.
Fractional exponents are just another way of writing roots. 91/2 = √9 = 3. 81/3 = ³√8 = 2. When the fraction is something like 2/3: a2/3 = (ⁿ√a)². So 82/3 = (³√8)² = 2² = 4. Our calculator handles all of this automatically.
Exponents with negative bases follow much the same rules as exponents with positive bases. If the exponent is an even positive integer, the values will be equal regardless of a positive or negative base. If the exponent is an odd positive integer, the result will have the same magnitude but will be negative.
While the rules for fractional exponents with negative bases are the same, they involve the use of imaginary numbers since it is not possible to take any root of a negative number. The calculator provided cannot compute imaginary numbers, and any inputs that result in an imaginary number will return "NAN."
Here's what trips people up the most. Watch out for these.
Mistake #1: Confusing (-2)⁴ with -2⁴. These are not the same. (-2)⁴ = (-2)×(-2)×(-2)×(-2) = 16. But -2⁴ = -(2×2×2×2) = -16. Parentheses matter.
Mistake #2: Thinking Negative Exponents Give Negative Answers. 2⁻³ is 1/8, not -8. A negative exponent means "one over," not "negative."
Mistake #3: Forgetting the Zero Exponent Rule. Anything (except 0) to the power of 0 is 1. 5⁰ = 1. 100⁰ = 1. Even (-999)⁰ = 1.
Mistake #4: Adding Exponents When Bases are Different. You can only use the product rule (adding exponents) when bases are the same. 2³ × 3⁴ can't be simplified — calculate separately: 8 × 81 = 648.
So when does this stuff actually matter? More often than you'd think.
That money in your savings account grows using exponents. A = P(1 + r)t. The "t" (time) is an exponent. That's why money grows faster the longer you leave it.
If a bacteria colony doubles every hour, after 10 hours it's 2¹⁰ = 1024 times bigger. That's exponential growth.
1 gigabyte = 1024 megabytes because computers work in base 2. 2¹⁰ = 1024. Exponents are everywhere in tech.
The Richter scale is logarithmic. A magnitude 5 quake is 10¹ = 10 times stronger than magnitude 4. A magnitude 6 is 10² = 100 times stronger.
The idea of exponents has been around for a long time. The ancient Egyptians and Babylonians used something similar. But the modern notation – that little number up there – was popularized by a French mathematician named René Descartes in the 1600s. Before that, people had to write out "multiply 2 by itself 4 times" every single time. Exponents were a game-changer for math.
Isaac Newton later used exponents to develop calculus. So yeah, this little notation is kind of a big deal.
It's because of the quotient rule. Think of 2³ ÷ 2³. That's 2⁽³⁻³⁾ = 2⁰. But 2³ ÷ 2³ is also 8 ÷ 8 = 1. So 2⁰ must equal 1. It's not a random rule — it's the only way math stays consistent.
Just type the base, then the exponent with a minus sign in front. For example, to calculate 2⁻³, type 2 as the base and -3 as the exponent. The calculator will give you 0.125, which is 1/8. Remember, a negative exponent doesn't mean a negative answer.
They look similar, but they're very different. 2³ means 2 × 2 × 2 = 8. 3² means 3 × 3 = 9. The base and the exponent are swapped, so the answer is different. Always pay attention to which number is the base and which is the exponent.