Square Root Calculator

How to use the square root calculator

1. Pick the root. Square root is the default. Choose cube root, or “Any root” to enter an index such as 4 for a fourth root.
2. Enter the number. Whole numbers, decimals and negatives all work. The examples load numbers from the textbook problems discussed below.
3. Read the results. The large figure is the root to 12 significant digits. The cards show the simplest radical form, the value rounded to two decimal places, the consecutive whole numbers it lies between, and a check.

The calculator opens on √72. Since 72 = 36 × 2 and √36 = 6, the simplest radical form is 6√2, about 8.485. That sits between 8 and 9, because 8² = 64 and 9² = 81.

What a square root is

A square root of a number m is a number whose square is m. OpenStax’s Elementary Algebra 2e (section 9.1) starts from 15² = 225: because 225 is the square of 15, 15 is a square root of 225. So is −15, since (−15)² = 225 as well. Every positive number has two square roots, one positive and one negative.

The radical sign √ means the principal square root, the positive one. √196 = 14, and −√196 = −14. The calculator shows the principal root, the way every calculator’s √ key does.

Zero has one square root, 0. A negative number has no real square root, because any real number squared is positive or zero. The textbook writes it plainly: √−169 is not a real number. In complex numbers it is 13i, which the calculator also shows for square roots.

Simplifying a square root

A square root is in simplest form when the number under the radical has no perfect square factor other than 1. To simplify, find the largest perfect square that divides the number, then take its root out in front, using √(ab) = √a × √b:

• √50 = √(25 × 2) = 5√2 (Example 9.12)
• √500 = √(100 × 5) = 10√5 (Example 9.13)
• √98 = √(49 × 2) = 7√2 (Intermediate Algebra 2e, Example 8.13)
• √72 = √(36 × 2) = 6√2

Using a smaller square works but leaves more to do. √72 = √(4 × 18) = 2√18, and 18 still contains 9, so it becomes 2 × 3√2 = 6√2. Starting with the largest square, 36, gets there in one step.

The calculator finds the largest square factor from the prime factorization. Each prime that appears twice comes out of the radical once; whatever is left over stays inside. 72 = 2³ × 3², so one pair of 2s and one pair of 3s come out as 2 × 3 = 6, and a single 2 stays inside.

Estimating a square root without a calculator

Most numbers are not perfect squares, so their square roots are irrational decimals that do not end or repeat. You can still bracket them quickly using the perfect squares around them. In Example 9.4, 60 lies between 49 and 64, so √60 lies between 7 and 8. Because 60 is much closer to 64, √60 is closer to 8; it is about 7.746.

When you need a decimal, round at the end. Example 9.5 rounds √17 = 4.1231056… to 4.12.

Cube roots and higher roots

The nth root of a number is the value that gives the number when raised to the nth power. Intermediate Algebra 2e (section 8.1) works out ∛64 = 4, ⁴√81 = 3 and ⁵√32 = 2.

The sign rules depend on whether the index is odd or even:

• Odd roots of negative numbers are negative: ∛−125 = −5, because (−5)³ = −125.
• Even roots of negative numbers are not real: no real number raised to the fourth power gives −16.

Higher roots simplify the same way, with perfect cubes or fourth powers instead of squares. ∛16 = ∛(8 × 2) = 2∛2, and ⁴√243 = ⁴√(81 × 3) = 3⁴√3. A root can also be written as a fractional exponent: ∛x = x^(1/3). The exponent calculator handles that form, including powers like 343^(2/3).

Where square roots show up

• Squares and areas: a square room of 144 square feet is √144 = 12 feet on a side.
• Right triangles: the Pythagorean theorem gives the long side as √(a² + b²). A TV that is 48 by 27 inches has a diagonal of √(2,304 + 729) ≈ 55.1 inches.
• Distance: the straight-line distance between two points on a graph is a square root, which the slope calculator also reports.
• Statistics: the standard deviation is the square root of the variance.
• Quadratic equations: the quadratic formula includes √(b² − 4ac), which is why exact answers often contain a radical.

Decimals, fractions and a common mistake

Square roots of decimals and fractions follow the same idea. √0.25 = 0.5 because 0.5 × 0.5 = 0.25, and √(9/16) = 3/4 because the root of a fraction is the root of the top over the root of the bottom. Watch the decimal point, though: √0.09 = 0.3, but √0.9 ≈ 0.9487, not 0.3. A decimal has an exact square root only when its digits form a perfect square and it has an even number of decimal places, as 0.09, 1.44 and 0.0144 (= 0.12²) do. Others, like 0.4, give a decimal that goes on without repeating: √0.4 ≈ 0.6325.

Notice also that the square root of a number between 0 and 1 is larger than the number itself: √0.25 = 0.5. Squaring makes such numbers smaller, so taking the root makes them larger.

The most common mistake is splitting a root across addition. Example 9.3 in the textbook makes the point with two calculations: √25 + √144 = 5 + 12 = 17, but √(25 + 144) = √169 = 13. The square root of a sum is not the sum of the square roots. Roots do split across multiplication, √(ab) = √a × √b, which is what makes simplifying possible.

Sources & assumptions

How this calculator gets its numbers. Full method: our methodology.

• The principal (non-negative) root is shown; even roots of negative numbers are not real, and for square roots the imaginary form is given.
• Odd roots of negative numbers are negative.
• Simplest radical form is computed for whole numbers up to 1 trillion by taking out the largest perfect nth-power factor.

Sources

1. OpenStax (Rice University), Elementary Algebra 2e, 9.1 Simplify and Use Square Roots. Square roots and the principal root (Example 9.1: √196 = 14), negative radicands (Example 9.2: √−169 is not a real number), estimating between whole numbers (Example 9.4: √60) and rounding (Example 9.5: √17 ≈ 4.12).
2. OpenStax (Rice University), Elementary Algebra 2e, 9.2 Simplify Square Roots. Simplifying with the largest perfect square factor: Example 9.12, √50 = 5√2; Example 9.13, √500 = 10√5.
3. OpenStax (Rice University), Intermediate Algebra 2e, 8.1 Simplify Expressions with Roots. nth roots: Example 8.3 (∛64 = 4, ⁴√81 = 3, ⁵√32 = 2), Example 8.4 (∛−125 = −5; ⁴√−16 is not real), Example 8.6 (∛49 ≈ 3.66, ⁴√51 ≈ 2.67).
4. OpenStax (Rice University), Intermediate Algebra 2e, 8.2 Simplify Radical Expressions. Product property of roots: Example 8.13, √98 = 7√2; Example 8.14, ∛16 = 2∛2 and ⁴√243 = 3⁴√3.

Last reviewed November 2, 2026. Decimal roots are rounded to 12 significant digits; perfect powers are shown exactly.