√ Root Calculator

Square, cube and nth roots — and why some have two answers

A root undoes a power. The nth root of a number is the value that, raised to the power n, gives that number back. The cube root of 2,744 is 14, because 14³ = 2,744. Written as an exponent it is the same operation: the nth root of x is x^1/n, which is why a calculator with only a power key can compute any root.

ⁿ√x = x^1/n

Higher roots pull numbers towards one, from either direction. The tenth root of a million is about 3.98; the tenth root of 0.000001 is about 0.251. Everything above one shrinks towards one and everything below one grows towards it, which is the behaviour that makes roots useful for averaging growth rates.

The two-answer problem

Both 5 and −5 square to 25, so the equation x² = 25 has two solutions. But the radical symbol √25 denotes only the positive one, 5, by convention — it is defined as a function and a function returns one value. This is why the quadratic formula carries an explicit plus-or-minus: the formula has to put back the second solution that the radical symbol deliberately excludes.

Odd roots are the simple case: every real number has exactly one real odd root, and the sign follows the input. The cube root of −8 is −2. Even roots of negatives have no real value at all, which is where imaginary numbers were invented to go — and why the quadratic formula calculator reports a negative discriminant as having no real roots.

Simplifying a surd

An irrational root is often better left exact than decimal. √72 becomes √(36 × 2) = 6√2, which is exact where 8.485 is rounded. The method is to pull out the largest perfect square factor; for cube roots, the largest perfect cube. This matters in exam answers and in any calculation where rounding errors would compound.

√(ab) = √a × √b  ·  √(a/b) = √a / √b

These two rules hold for non-negative values and fail for negatives. Applying √(ab) = √a × √b to √(−4 × −9) gives 6 correctly, but √−4 × √−9 gives −6 if you reason with imaginary units carelessly. Keep the rule to non-negative inputs.

Roots as averages of growth

To find the average annual growth rate behind a total change, you take a root rather than an arithmetic mean. An investment that grows 50% one year and falls 50% the next has an arithmetic mean of zero and an actual outcome of −25%: 1.5 × 0.5 = 0.75. The correct average is the geometric mean, √(1.5 × 0.5) = 0.866, a loss of 13.4% a year, which compounds to exactly the right answer.

That is the same calculation as an annualised return, which is why the ROI calculator raises the total growth to the power of one over the number of years. Any time a rate is applied repeatedly, the honest average is a root and not a sum divided by n.

Why √2 is irrational

Suppose √2 were a fraction a/b in lowest terms. Then a² = 2b², so a² is even, so a is even. Write a = 2k: then 4k² = 2b², so b² = 2k², so b is even too. But a and b were in lowest terms and cannot both be even. The assumption fails, so no such fraction exists. The same argument works for the square root of any non-square integer, and it is the oldest proof by contradiction that survives.

A worked example

A cubical tank holds 2,744 litres. Since a litre is a cubic decimetre, the side is the cube root of 2,744 decimetres: 14 dm, or 1.4 metres. To double the volume you do not double the side — you multiply it by the cube root of 2, about 1.26, giving 1.76 m. This is the scaling rule that makes large objects disproportionately heavy: volume grows as the cube of the length, so a tank twice as tall holds eight times as much. The volume calculator runs the forward direction.

Once you have this figure, the exponent calculator is where it usually goes next. the log calculator is the related one people reach for afterwards.

Frequently Asked Questions (FAQs)