Exponent & Root Calculator
Calculate powers (aᵇ) and roots (√, ∛, any n-th root) with exact big-integer results for whole-number powers.
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Power: aᵇ
Root: ⁿ√x
How Powers and Roots Work
Four rules cover almost everything: multiplying powers with the same base adds the exponents (aᵐ × aⁿ = aᵐ⁺ⁿ), raising a power to another power multiplies the exponents ((aᵐ)ⁿ = aᵐⁿ), a negative exponent flips the number into a fraction (a⁻ⁿ = 1/aⁿ), and any non-zero number raised to the power of zero equals 1 (a⁰ = 1). The n-th root is simply a fractional exponent in disguise: ⁿ√a = a^(1/n) — square root is a^(1/2), cube root is a^(1/3), and so on. This equivalence is exactly how calculators (and this tool) compute roots internally: instead of a separate root algorithm, they raise the number to a fractional power.
Concrete example: 2³ × 2⁴ = 2⁷ = 128 (add exponents: 3+4=7) rather than multiplying 8×16 by hand. Or (3²)³ = 3⁶ = 729 (multiply exponents: 2×3=6). And ∛27 = 27^(1/3) = 3, because 3³ = 27 — the root and the cube undo each other exactly, showing why fractional exponents and roots are two notations for the same operation.
What's Worth Knowing
- Negative exponents mean reciprocal, not negative result: 2⁻³ = 1/2³ = 1/8 = 0.125 — never a negative number (unless the base itself is negative and the exponent is odd).
- Zero exponent is always 1: for any a ≠ 0, a⁰ = 1, including large numbers like 1,000,000⁰ = 1. Only 0⁰ is left undefined.
- Even roots of negative numbers have no real answer: √-4 doesn't exist in real numbers, since no real number squared gives a negative result — but odd roots of negatives work fine, ∛-8 = -2.
- Big-integer precision matters for whole-number powers: 2¹⁰⁰ has 31 exact digits; ordinary floating-point math would round this, but big-integer arithmetic keeps every digit exact.
- Fractional exponents combine a power and a root: a^(m/n) means taking the n-th root of a first and then raising it to the m-th power (or doing it in the opposite order for the same result) — for example, 8^(2/3) = (∛8)² = 2² = 4.
Frequently Asked Questions
Why does a negative exponent give a fraction instead of a negative number?
A negative exponent means "take the reciprocal": a⁻ⁿ = 1 ÷ aⁿ. So 2⁻³ isn't -8, it's 1/2³ = 1/8 = 0.125. The result only becomes negative if the base itself is negative and the exponent is odd.
Why does any number to the power of zero equal 1?
It follows directly from the division-of-powers rule: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and any number divided by itself equals 1. This holds for every non-zero base — only 0⁰ is left mathematically undefined.
How is taking a root the same thing as raising to a power?
The n-th root of a number is defined as that number raised to the power 1/n: ⁿ√a = a^(1/n). A square root is a^(1/2), a cube root is a^(1/3) — that's why this tool (and every calculator) computes roots using the same exponentiation engine instead of a separate method.
Can I take the root of a negative number?
It depends on whether the root is odd or even. Odd roots of negative numbers work fine — ∛-8 = -2, because (-2)³ = -8. Even roots of negative numbers (square root, 4th root, etc.) have no real result, since no real number squared or raised to an even power gives a negative value; the tool flags these cases.
Why does 2¹⁰⁰ show all its digits instead of scientific notation like 1.27E+30?
For whole-number exponents, the tool uses big-integer arithmetic rather than standard floating-point math, so it can track every digit exactly with no rounding — 2¹⁰⁰ = 1,267,650,600,228,229,401,496,703,205,376, all 31 digits precise. Roots and fractional powers, which can't always be expressed as exact integers, fall back to high-precision decimal calculation instead.
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