Exponent Calculator
Enter a base and an exponent to get aⁿ, its scientific notation and, for whole-number results, every digit. Negative, decimal and fractional exponents (0.5 = square root) work too; very large results are written as powers of ten.
- Expanded2¹⁰
- Scientific notation1.024 × 10³
- Exact value1,024
How it is calculated
A power is shorthand for multiplying a number by itself repeatedly: in aⁿ, a is the base and n the exponent. The rules extend to negative and fractional exponents: 2⁻¹ = 1/2, 9^(1/2) = √9 = 3. With a negative base, an even or odd exponent sets the sign: (−2)⁴ = 16, (−2)³ = −8. A negative base has a real fractional power only when the fraction’s denominator is odd: (−8)^(1/3) = −2, but (−8)^(1/2) is not real. The calculator writes whole-number results out in full up to 60 digits and gives larger ones in scientific notation with their digit count.
aⁿ = a × a × … × a · a⁻ⁿ = 1 ÷ aⁿ · a^(p/q) = q√(aᵖ) · a⁰ = 1a: base · n: exponent (power). A positive whole exponent multiplies the base by itself n times; a negative one gives the reciprocal and a fractional one a root.- Product: aᵐ × aⁿ = aᵐ⁺ⁿ (2³ × 2⁴ = 2⁷ = 128).
- Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (5⁶ ÷ 5⁴ = 5² = 25).
- Power of a power: (aᵐ)ⁿ = aᵐˣⁿ ((3²)³ = 3⁶ = 729).
- Power of a product: (a × b)ⁿ = aⁿ × bⁿ (6² = 2² × 3² = 36).
- Negative and fractional exponents: a⁻ⁿ = 1 ÷ aⁿ, a^(1/n) = ⁿ√a.
Example
3⁵ = 3 × 3 × 3 × 3 × 3 = 243. 2⁻³ = 1 ÷ 2³ = 1/8 = 0.125. 8^(2/3) = ∛(8²) = ∛64 = 4. 2¹⁰⁰ = 1,267,650,600,228,229,401,496,703,205,376, a 31-digit number, about 1.26765 × 10³⁰ in scientific notation.
Frequently asked questions
Why is any number to the power zero equal to 1?
From the quotient rule: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and a number divided by itself is 1 (a ≠ 0). For example 5³ ÷ 5³ = 125 ÷ 125 = 1 = 5⁰.
What does a negative exponent mean?
It is the reciprocal of the positive power: a⁻ⁿ = 1 ÷ aⁿ. 10⁻³ = 1 ÷ 1,000 = 0.001; 2⁻³ = 1/8 = 0.125. Zero to a negative power is undefined, as it would divide by zero.
Is a fractional exponent the same as a root?
Yes: a^(1/n) = ⁿ√a, and a^(p/q) is the q-th root of aᵖ. 27^(1/3) = ∛27 = 3 and 16^(3/4) = (∜16)³ = 2³ = 8. The root calculator does roots directly.
Are −2⁴ and (−2)⁴ the same?
No. The exponent binds before the minus sign: −2⁴ = −(2⁴) = −16, but (−2)⁴ = 16. Use brackets for a negative base; entering −2 as the base here means (−2)⁴.
What is 0⁰?
Algebra and combinatorics take 0⁰ = 1; IEEE 754-2019 (§9.2.1, pow) and JavaScript (ECMA-262 Number::exponentiate) also return 1 for a zero exponent, and so does this calculator. In analysis, though, x⁰ and 0ˣ tend to different limits, so 0⁰ is an indeterminate form.
What is scientific notation?
Writing a number as one between 1 and 10 times a power of ten: 2¹⁰⁰ ≈ 1.26765 × 10³⁰, 0.000052 = 5.2 × 10⁻⁵. The exponent says how many places the decimal point moves.