A negative exponent denotes the reciprocal of the positive power: a⁻ⁿ = 1 ÷ aⁿ. For example, 2⁻³ = 1 ÷ 2³ = 1 ÷ 8 = 0.125.
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Powers with any exponent and roots of any index, with steps.
A power is the repeated multiplication of a base by itself; the exponent says how many times it repeats, so aⁿ = a × a × … × a (n factors). This calculator accepts positive integer exponents, negative exponents, which mean the reciprocal, a⁻ⁿ = 1 ÷ aⁿ, and fractional exponents, which equal roots, because a^(1/n) is the nth root of a. In root mode it solves the inverse path: given the radicand and the index n, it computes a^(1/n), the square root when n = 2, the cube root when n = 3, and so on. The tool handles the delicate cases carefully: a negative base with a fractional exponent has no real result (flagged as complex), the even-index root of a negative number does not exist in the reals, but the odd-index root of a negative does have a real negative result (the cube root of −27 is −3). Every answer comes with didactic steps, and in root mode a check that raises the result back to the index so you can verify it. Everything runs locally in your browser with 64-bit floating point, displaying up to 12 significant figures.
A power aⁿ is just repeated multiplication of the base: 2⁵ means 2 × 2 × 2 × 2 × 2. When the exponent is negative it denotes the reciprocal, a⁻ⁿ = 1 ÷ aⁿ, and when it is fractional it becomes a root, because a^(1/n) is by definition the number that, raised to n, gives back a. Exponentiation and roots are therefore inverse operations: cubing and taking the cube root cancel each other out.
In root mode the tool computes a^(1/n). For a positive radicand this is straightforward; for a negative radicand it distinguishes the index: if n is even there is no real root (no real number squared is negative), but if n is odd the result is real and negative, obtained as −(|a|^(1/n)).
Example 1, power with a negative exponent: 2⁻³. By the reciprocal rule, 2⁻³ = 1 ÷ 2³ = 1 ÷ 8 = 0.125. The calculator shows the base and exponent first, then the step 1 ÷ 2³, and finally the decimal value 0.125.
Example 2, even-index root: the fourth root of 81. In root mode, radicand 81 and index 4 give 81^(1/4) = 3, because 3⁴ = 3 × 3 × 3 × 3 = 81. The check raises the result back: 3⁴ = 81, confirming the answer.
A few cases deserve attention. The square root of a negative number has no real value and the tool warns you; a negative base with a fractional exponent hits the same wall (a complex result). By convention 0⁰ returns 1, but 0 raised to a negative exponent is undefined, since it would require dividing by zero. And be careful typing fractional exponents as decimals: 1/3 becomes 0.3333…, so for an exact cube root prefer root mode with index 3.
Powers and roots appear across applied math: compound interest and growth use aⁿ, standard deviation and the distance between points use the square root, scientific notation scales by powers of 10, and areas and volumes involve squares and cubes.
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A negative exponent denotes the reciprocal of the positive power: a⁻ⁿ = 1 ÷ aⁿ. For example, 2⁻³ = 1 ÷ 2³ = 1 ÷ 8 = 0.125.
Calculations run in your browser. No data is sent to a server.