Square Roots, Cube Roots and nth Roots

A root is the inverse operation of a power: the nth root of a number a is a number that gives a when raised to the power n. In brief: √49 = 7 because 7² = 49, ∛64 = 4 because 4³ = 64, and ⁴√81 = 3 because 3⁴ = 81.

Square roots appear in geometry and the Pythagorean theorem, cube roots in volume problems, and higher roots in growth rates and science. This guide explains the definitions, shows how to find roots of perfect powers, how to estimate roots that are not whole numbers and which rules govern negative numbers.

What is a square root?

The square root of a nonnegative number a is the nonnegative number whose square is a. It is written √a. Because 6² = 36, we have √36 = 6. The equation x² = 36 has two solutions, 6 and −6, but the symbol √36 means only the nonnegative one, 6.

The number under the root sign is the radicand. A square root of a negative number is not a real number, because no real number squared gives a negative result. Such roots lead to complex numbers.

The square root of a square is the absolute value: √(x²) = |x|. For example, √((−5)²) = √25 = 5, not −5.

What are cube roots and nth roots?

The cube root of a, written ∛a, is the number whose cube is a. Unlike square roots, cube roots exist for negative numbers: ∛(−27) = −3, because (−3)³ = −27. Each real number has exactly one real cube root.

The nth root of a is written ⁿ√a, where n is the index. It is the number b with bⁿ = a. For even n, the radicand must be nonnegative and the root is taken to be nonnegative. For odd n, the radicand can be any real number, and the root has the same sign as the radicand.

Roots are also fractional exponents: ⁿ√a = a^1/n.

How do you estimate a root that is not a whole number?

1. Find the perfect powers just below and just above the radicand.
2. The root lies between their roots.
3. Test a decimal between them and refine.

Example: √50

Since 49 < 50 < 64, we get 7 < √50 < 8. The number 50 is close to 49, so the root is slightly above 7. Testing 7.1 gives 7.1² = 50.41, which is a bit too large. Testing 7.07 gives about 49.98. So √50 ≈ 7.07.

Example: ∛20

Since 2³ = 8 and 3³ = 27, ∛20 lies between 2 and 3. Because 2.7³ = 19.683 and 2.8³ = 21.952, the cube root is a little above 2.7: ∛20 ≈ 2.71.

Example: exact root

√(144/25) = 12/5 = 2.4, because the root of a fraction is the root of the numerator over the root of the denominator.

Perfect squares and square roots table

Perfect cubes and cube roots table

Where does the root symbol come from?

Ancient Babylonian clay tablets contain remarkably accurate approximations of √2, and Greek geometers worked with square roots as side lengths of squares. The radical sign √ is usually thought to have developed from a stylised letter r, from the Latin radix meaning root, and it appeared in print in the sixteenth century. The vinculum, the bar over the radicand, was added later, and the index for higher roots became common in the seventeenth century.

Where are roots used today?

The side of a square with area 81 m² is √81 = 9 m. The edge of a cube with volume 125 cm³ is ∛125 = 5 cm. The Pythagorean theorem requires square roots to get lengths, and the standard deviation in statistics is a square root of the variance. Average annual growth over n years is found with an nth root: if a value triples in 4 years, the yearly factor is ⁴√3 ≈ 1.316, about 31.6% per year.

Which mistakes are common with roots?

Splitting a root over a sum

√(9 + 16) = √25 = 5, but √9 + √16 = 7. Roots do not distribute over addition.

Forgetting the second solution

The equation x² = 25 has two solutions, ±5, even though √25 = 5.

Thinking √(x²) = x

The correct statement is √(x²) = |x|. For x = −3 the root is 3.

Taking an even root of a negative number

√(−16) is not a real number. The cube root ∛(−8) = −2 is fine because the index is odd.