Exponent and Root Calculator

How to use the exponent and root calculator

1. Choose Power (xⁿ) or Root (ⁿ√x).
2. Type the Base x. Whole numbers, decimals and fractions work: 2, -8, 1.5, 3/4.
3. For a power, type the Exponent n: 10, -3, 0.5 or 2/3. For a root, type the Root index n (2 = square root, 3 = cube root).
4. The Result updates as you type; Calculate does the same and Copy copies the exact form.

Under the result you get the decimal value, and the Step by step panel explains each rule used: negative exponent, fraction exponent, prime factorization, grouping of factors, imaginary unit.

Exact results, not just decimals

A calculator shows √72 ≈ 8.485281374. This tool also gives the exact simplified radical, 6√2, the form expected in algebra homework. It works like the method taught in class:

1. Factor the number into primes: 72 = 2³ × 3².
2. For a square root, each pair of equal factors leaves the root as one factor: 2² and 3² give 2 × 3 = 6.
3. What is left inside stays under the root: 2. So √72 = 6√2.

For a cube root you take out groups of three (∛54 = ∛(2 × 3³) = 3∛2), for a fourth root groups of four, and so on. A fraction under a root is rationalized, so √(1/2) is written √2/2.

Whole-number powers are exact too, however large: 2¹⁰⁰⁰ is written out with all its 302 digits, and fractions stay fractions ((3/4)² = 9/16, 2⁻³ = 1/8). Results with more than 50,000 digits are given in scientific notation instead.

The rules of exponents

• Positive whole exponent: repeated multiplication, 3⁴ = 3 × 3 × 3 × 3 = 81.
• Zero exponent: x⁰ = 1 for any nonzero x. 0⁰ is taken as 1, the usual convention in algebra; 0 to a negative power is undefined.
• Negative exponent: x⁻ⁿ = 1 / xⁿ, so 10⁻² = 1/100 = 0.01.
• Fractional exponent: x^(p/q) = q-th root of xᵖ. So 8^(2/3) = ∛(8²) = ∛64 = 4 and 16^0.25 = ∜16 = 2. A decimal exponent is first turned into a fraction (0.25 = 1/4).

Roots of negative numbers

An odd root of a negative number is a real, negative number: ∛(−8) = −2, because (−2)³ = −8.

An even root of a negative number has no real value, since any real number squared is positive or zero. It exists among complex numbers, using the imaginary unit i = √−1: √−16 = √16 × √−1 = 4i. For square roots the tool writes this exact form (√−8 = 2√2i). For other even roots, such as ∜(−16), it gives the principal complex value in decimal form, 1.414 + 1.414i, and explains it in the steps.