Arithmetic
Exponent Calculator
Raise any base to any power and watch the result update as you type. Positive, negative, fractional, and decimal exponents are all handled — with every step of the working shown.
Raise a base to a power
bⁿ210
Result · bⁿ
=1,024
2 raised to the power of 10
Positive whole power
base b
2
exponent n
10
scientific
1.02e+3
Updates live
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The basics
What is an exponent?
An exponent is a shorthand for repeated multiplication. Instead of writing a number out many times, you write the number once — the base — with a small raised number — the exponent — telling you how many copies to multiply together. It is one of the most reused ideas in all of math, powering everything from area and volume to compound interest and scientific notation.
bⁿ = b × b × … (n times)
Base and exponent
The base is the number being multiplied; the exponent counts how many times. Read b to the n as “b raised to the power of n.”
2³ = 2 × 2 × 2 = 8
b² = square · b³ = cube
Squares and cubes
The exponent 2 is called squaring — the area of a square of side b. The exponent 3 is cubing — the volume of a cube of side b.
5² = 25 · 5³ = 125
Method
How to calculate an exponent by hand
For a positive whole-number power, the whole job is one repeated multiplication. Other kinds of exponent reduce to this after one conversion.
- 1
Write the base out n times
Lay out the base b as many times as the exponent n says, joined by multiplication.
4³ → 4 × 4 × 4 - 2
Multiply left to right
Combine two factors at a time, carrying the running product forward to the next factor.
4 × 4 = 16, then 16 × 4 = 64 - 3
Handle special exponents first
If n is negative, flip to 1 ÷ bⁿ. If it is a fraction, take the matching root. Then evaluate the whole-number part as above.
10⁻² = 1 ÷ 10² = 0.01
Every kind
The six kinds of exponent
The exponent looks the same wherever it appears, but its meaning shifts with its sign and form. Here is every case this calculator recognizes.
b³ = b × b × b
Positive whole
Multiply the base by itself that many times.
2⁴ = 16
b¹ = b
Power of one
Any number to the first power is itself.
9¹ = 9
b⁰ = 1
Power of zero
Any non-zero base to the zero power is one.
5⁰ = 1
b⁻ⁿ = 1 ÷ bⁿ
Negative
Flip to the reciprocal of the positive power.
2⁻² = 0.25
b^(1/m) = ᵐ√b
Fractional
The denominator becomes the root of the base.
8^(1/3) = 2
b^2.5 = b² · √b
Decimal
Split a decimal power into a whole part and a root.
4^1.5 = 8
Worked examples
See it across exponent types
The same base, four different exponents — each worked the way its type demands. Switch tabs to compare.
Given 2⁵ — a whole-number power
Multiply the base by itself 5 times
2 × 2 × 2 × 2 × 2
= 4 × 2 × 2 × 2 = 8 × 2 × 2 = 16 × 2
2⁵ = 32
The laws
Rules of exponents worth remembering
A handful of identities let you combine and simplify powers without expanding them. These are the laws of exponents — the backbone of algebra.
bᵐ × bⁿ = bᵐ⁺ⁿ
Product rule
Multiplying powers of the same base adds the exponents.
2³ × 2² = 2⁵ = 32
bᵐ ÷ bⁿ = bᵐ⁻ⁿ
Quotient rule
Dividing powers of the same base subtracts the exponents.
2⁵ ÷ 2² = 2³ = 8
(bᵐ)ⁿ = bᵐⁿ
Power of a power
Raising a power to another power multiplies the exponents.
(2³)² = 2⁶ = 64
(ab)ⁿ = aⁿbⁿ
Power of a product
A product raised to a power distributes over each factor.
(2·3)² = 4 · 9 = 36
b⁰ = 1
Zero exponent
Any non-zero base to the power zero is one — it follows from the quotient rule.
99⁰ = 1
b⁻ⁿ = 1 ÷ bⁿ
Negative exponent
A negative exponent is the reciprocal of the positive power.
2⁻³ = 0.125
Quick chart
What is b to the power of n?
Pick a base and read off every power from b¹ to b¹² at a glance — the answer to hundreds of “2 to the power of 8” style searches in one place.
Powers of
21
2
22
4
23
8
24
16
25
32
26
64
27
128
28
256
29
512
210
1,024
211
2,048
212
4,096
2¹ through 212 · square = 2², cube = 2³
Reference
Squares and cubes, 1 to 15
The most-used powers in school math: every number from 1 to 15 squared and cubed. Worth committing to memory.
n
n² (square)
n³ (cube)
1
1
1
2
4
8
3
9
27
4
16
64
5
25
125
6
36
216
7
49
343
8
64
512
9
81
729
10
100
1000
11
121
1331
12
144
1728
13
169
2197
14
196
2744
15
225
3375
Where it shows up
Why exponents matter
Exponents are not a classroom curiosity — they describe growth, scale, and measurement across science and finance.
A = P(1 + r)ⁿ
Compound interest
Money grows by the same factor each period, so the number of periods sits in the exponent — small rates compound into large sums.
m × 10ⁿ
Scientific notation
Powers of ten let scientists write enormous and tiny numbers compactly, from the mass of a star to the size of an atom.
2ⁿ
Computing
Bits double the number of states with each added position, so memory, addresses, and storage are all powers of two.
Watch out
Common mistakes with exponents
A few errors trip people up again and again. Knowing them is half the battle.
2³ ≠ 2 × 3
It is not multiplication
An exponent is repeated multiplication, not the base times the exponent. The raised number counts factors — it is not a factor itself.
−3² ≠ (−3)²
Mind the minus sign
Without parentheses the exponent binds tighter than the minus, so the negative is applied after squaring. Parentheses change the answer — the order of operations calculator shows exactly where a power sits in the precedence ladder.
b⁰ = 1, not 0
Zero power is one
A base to the zero power is one, never zero. The exponent counting “no factors” leaves the multiplicative identity behind.
(a + b)² ≠ a² + b²
Powers do not distribute over sums
You cannot square a sum term by term. Expanding correctly leaves a cross term that is easy to forget.
FAQ
Common questions
An exponent tells you how many times to multiply a base by itself. In bⁿ, b is the base and n is the exponent, so 2³ means 2 × 2 × 2 = 8. The exponent is the small raised number to the right of the base — also called a power or index.
A negative exponent means the reciprocal of the positive power: b⁻ⁿ = 1 ÷ bⁿ. For example, 2⁻³ = 1 ÷ 2³ = 1 ÷ 8 = 0.125. The base moves to the denominator and the exponent becomes positive.
A fractional exponent is a root. The denominator is the root and the numerator is the power: b^(1/m) is the m-th root of b, and b^(p/m) is the m-th root of bᵖ. So 9^(1/2) is the square root of 9, which is 3, and 8^(1/3) is the cube root of 8, which is 2.
Any non-zero number raised to the power of 0 equals 1. This keeps the laws of exponents consistent — dividing bⁿ by bⁿ gives b⁰, and any number divided by itself is 1. (0⁰ is left undefined or treated as 1 by convention.)
For a positive whole-number exponent, write the base out that many times and multiply. For 4³, write 4 × 4 × 4 and multiply left to right: 4 × 4 = 16, then 16 × 4 = 64. For negative or fractional exponents, first convert to a reciprocal or a root, then evaluate.
They are closely related. The exponent is the raised number itself (the n in bⁿ), while the power usually refers to the whole expression bⁿ or its result. People say "2 to the power of 3" and "an exponent of 3" to mean the same thing.
Yes. A decimal base like 1.5² is fine: 1.5 × 1.5 = 2.25. A decimal exponent is treated as a fractional power, so 2^2.5 = 2² × 2^0.5 = 4 × √2 ≈ 5.657. The calculator handles decimals in both fields.
A fractional exponent asks for a root. Even roots of negative numbers — like the square root of −4 — have no real value because no real number squared is negative. The result is a complex number, which sits outside the real-number line this calculator works on.
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Written by
UC
Author
Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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