How this calculator works
- Step 1Whole numberbⁿMultiply the base by itself n times.
- Step 2Negativeb⁻ⁿ = 1 ÷ bⁿTake the reciprocal of the positive power.
- Step 3Fractionb^(1/q)Take the q-th root, then any whole power.
An exponent says how many times a base is used as a factor. The calculator works for whole, negative and decimal exponents.
A negative base only has a real answer for a whole-number exponent, so the calculator says so instead of showing an error value.
Worked example: 2 to the power 10
- Power
bⁿ = b × b × … × b (n factors)
- bBase 2
- nExponent 10
- 2⁵2 × 2 × 2 × 2 × 232
- 2¹⁰2⁵ × 2⁵ = 32 × 321,024
Splitting 2¹⁰ into 2⁵ × 2⁵ uses the product rule, bᵐ × bⁿ = bᵐ⁺ⁿ, and saves multiplications.
Negative and fractional exponents
- 10⁻³0.001One over 10³, which is one over a thousand.
- 16^0.54The square root of 16.
- (−2)³−8An odd power keeps the minus sign.
Read the result
Exponents grow quickly: 1.05¹⁰ is 1.628895, so ten 5% steps add about 63%, not 50%.
The reciprocal beneath the answer is b⁻ⁿ, the same power with the sign of the exponent flipped.
Assumptions and limits
- Real numbersNo complex answers for negative bases.
- 0 to a negative powerUndefined, because it divides by 0.
- 0⁰ = 1The usual convention in algebra.
- DisplaySix decimals, or scientific notation for huge or tiny answers.
Related powers and roots
Powers, roots and logarithms are three views of the same relationship.
To undo a power, use the root calculator for n-th roots or the log calculator to find the exponent itself. For very large or small results, the scientific notation calculator writes them as a × 10ⁿ.
Common mistakes
Mistake: Reading 2¹⁰ as 2 × 10.
Instead: Multiply 2 by itself ten times.
Mistake: Treating −2² as (−2)².
Instead: Without brackets, −2² means −(2²) = −4.
Mistake: Thinking a negative exponent gives a negative answer.
Instead: It gives the reciprocal: 2⁻¹ = 0.5.
Checking the method
Every golden case is re-worked by exact repeated multiplication, or by Newton's method for roots.
An independent script works with exact fractions for whole exponents and 40-digit roots for fractional ones, then compares each answer.
Read how we test our calculators, or browse more maths calculators.