Powers and roots
An exponent says how many times to multiply a number by itself: 2¹⁰ = 2 × 2 × … × 2 (ten times) = 1,024. Roots run the other way: the cube root of 27 is the number that, multiplied by itself three times, gives 27. Exponents describe compound growth, area and volume scaling, computer storage (2¹⁰ bytes = 1 KiB) and scientific notation.
Exponent rules
bᵐ × bⁿ = bᵐ⁺ⁿ bᵐ / bⁿ = bᵐ⁻ⁿ (bᵐ)ⁿ = bᵐⁿ b⁰ = 1 b⁻ⁿ = 1 / bⁿ b^(1/n) = ⁿ√b b^(m/n) = (ⁿ√b)ᵐ Simplest radical: ⁿ√x = k · ⁿ√m, where kⁿ is the largest perfect n-th power dividing x
Worked examples
3⁴ = 3 × 3 × 3 × 3 = 81. 10⁻² = 1 / 100 = 0.01.
√72: the largest perfect square dividing 72 is 36, so √72 = √36 × √2 = 6√2 ≈ 8.485281. ∛54 = ∛27 × ∛2 = 3∛2 ≈ 3.779763.
Notes
- Results use double-precision arithmetic (about 15–16 significant digits). Results beyond about 1.8 × 10³⁰⁸ show as ∞.
- Whole-number roots of perfect powers (∛27 = 3) are shown exactly, without rounding noise.
- Simplest radical form is computed for whole numbers up to 10¹².
Sources: OpenStax, College Algebra 2e, §1.2 Exponents and Scientific Notation and §1.3 Radicals and Rational Exponents.
Frequently asked questions
What does a negative exponent mean?
A negative exponent means "one over": b⁻ⁿ = 1 / bⁿ. So 2⁻³ = 1/8 = 0.125. It follows from the rule bᵐ / bⁿ = bᵐ⁻ⁿ: 2² / 2⁵ = 2⁻³.
What is a fractional exponent?
A fractional exponent is a root. b^(1/n) is the n-th root of b, and b^(m/n) is the n-th root of b raised to the m: 8^(2/3) = (∛8)² = 2² = 4. Enter 2/3 as 0.6666… or use root mode with index 3 and then square.
Why is 0⁰ shown as 1?
By convention in algebra, combinatorics and most programming languages, 0⁰ = 1 (it is the empty product, and it keeps formulas such as the binomial theorem valid). In calculus, the limit form 0⁰ is indeterminate — that is a different question.
Can I take the root of a negative number?
Odd roots, yes: ∛(−27) = −3, because (−3)³ = −27. Even roots of negative numbers (√−4) are not real numbers — they are imaginary (2i) — so the calculator reports that instead of a misleading value.