Root Calculator

How the Root Calculator works

A root undoes a power: the cube root of 27 is 3 because 3³ = 27. Most roots are irrational, so they never end or repeat, which is why maths keeps them as a simplified radical rather than a rounded decimal. Simplifying pulls perfect powers out from under the root sign. That exact form is what homework, exams and geometry answers usually expect.

How the root calculator works

The formula is the nth root of x = x^(1/n). To simplify, the calculator factors x into primes and moves every complete group of n identical primes outside the root. For √72, 72 = 2³ × 3², so one 2 and one 3 come out, leaving 6√2. The decimal value is then raised to the nth power as a check.

First example. √72 = 6√2 ≈ 8.485281, because 72 = 36 × 2 and √36 = 6.

Second example. ∛27 = 3 exactly, since 3 × 3 × 3 = 27. The calculator marks it as a perfect cube.

How many square roots does a number have? Every positive number has two, such as +5 and -5 for 25; the √ symbol means the positive one, called the principal root.

Squares, square roots, simplified radicals, cubes and cube roots for 1 to 25

Tips and pitfalls

• Even roots of negatives. √-16 is 4i, not a real number; odd roots like ∛-8 = -2 are fine.
• Do not split sums. √(9 + 16) is 5, not √9 + √16 = 7.
• Simplify fully. √48 = 4√3, not 2√12; take out the largest perfect square.
• Roots are fractional powers. The fourth root of x is x^0.25, useful in spreadsheets.

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