Square Root Calculator — Square Roots, Cube Roots & Nth Roots

What Is a Square Root Calculator?

A square root calculator finds the number that, multiplied by itself, gives your input. This free tool also finds cube roots (∛27 = 3), any nth root (⁴√81 = 3), and simplifies radicals to exact form (√50 = 5√2) — every result with step-by-step working and a decimal approximation.

How to Use the Square Root Calculator

The tool has four modes so you can start with the information you already know.

1. Choose a mode that matches your question.
2. Enter your number (and n for Nth Root mode).
3. Select Calculate to generate the result.
4. Check the result cards for the exact answer, the decimal, the step-by-step working and a verification you can check by hand.

Every result shows the exact form, the step-by-step working, and a decimal approximation rounded to four places — so you can check homework by hand and still see the precise value.

What Is a Square Root?

The square root of a number x is the value that, multiplied by itself, gives x. If y² = x, then y = √x. Squaring "undoes" a square root, and a square root "undoes" squaring.

A square root is also written as a fractional exponent: √x = x^(1/2). So √144 = 144^(1/2) = 12. This notation connects roots to the exponent rules you already know — for example, (√5)² = 5.

Perfect Squares Worth Memorizing

Notice the pattern: 10² = 100 and 11² = 121, so √110 must sit between 10 and 11 (about 10.4881). Bounding an unknown root between two perfect squares is the fastest way to estimate one by hand.

What Is a Cube Root?

The cube root of a number x is the value that, multiplied by itself twice (three copies total), gives x. If y³ = x, then y = ∛x. Unlike square roots, cube roots work for negative numbers too, because three negatives multiply to a negative.

In exponent form: ∛x = x^(1/3). So ∛27 = 27^(1/3) = 3.

Perfect Cubes Worth Memorizing

How to Simplify Radicals, Step by Step

Some roots don't come out even — √50 is not a whole number. But you can still write it in exact "simplified radical form" by pulling perfect powers out of the root. Follow these four steps.

Step 1: Find the largest perfect square that divides the number

For 50, the perfect-square divisors are 1 and 25. The largest is 25.

Step 2: Split the radicand

Step 3: Root the perfect square and move it outside

Step 4: Add the decimal approximation

The rule is always the same: group the prime factors into pairs for square roots (or triples for cube roots), move one copy of each complete group outside, and leave the leftovers inside.

More Worked Examples

Square Root vs Cube Root

The two operations look alike but behave differently — the comparison below settles the most-tested distinctions.

Nth Roots at a Glance

The nth root generalizes square and cube roots: ⁿ√x is the value that gives x when multiplied by itself n times.

For even n, x must be 0 or positive; for odd n, negative x gives a negative result. Match the root's index to the equation's exponent: x⁴ = 81 → x = ±⁴√81 = ±3.

Where Are Roots Used?

For example, a right triangle with legs 3 and 4 has hypotenuse c = √(3² + 4²) = √25 = 5.

Square Root & Cube Root Examples

How to Estimate a Root by Hand

1. Bound it. Find the two perfect powers it sits between — e.g. 7² = 49 < 50 < 64 = 8², so √50 is just above 7.
2. Guess and square. Try 7.07: 7.07² = 49.9849, a hair under 50.
3. Refine once. Try 7.071: 7.071² = 49.999041 — essentially 50.
4. Round last. √50 ≈ 7.0711. Keep the exact form (5√2) through your working and round only the final answer.
5. Check with the inverse. Squaring (or cubing) your answer must return the original number.

Common Mistakes With Square Roots

1. The 21 trap: thinking √144 + ∛27 = 21

This is the most popular wrong answer on the internet. The trap is a single wrong step: reading ∛27 as 9 instead of 3. Check it: 3 × 3 × 3 = 27, so ∛27 = 3. The real sum is 12 + 3 = 15, not 12 + 9 = 21.

2. Splitting a root across a sum: √(a + b) ≠ √a + √b

A classic algebra error. Take a = 16, b = 9: √(16 + 9) = √25 = 5, but √16 + √9 = 4 + 3 = 7. Not equal. You can split a root across multiplication — √(a × b) = √a × √b — but never across addition.

3. Taking an even root of a negative number

√(−4) is not a real number, because no real value squared gives −4. Odd roots are fine with negatives: ∛(−8) = −2.

4. Forgetting that every positive number has two square roots

x² = 144 has two solutions: x = 12 and x = −12. The √ symbol means the principal (non-negative) root, so √144 = 12 — but if you're solving an equation, don't drop the ±.

5. Rounding too early

Approximating √2 as 1.41 and then multiplying by 5 gives 7.05, not the true 7.0711. Keep the exact form (5√2) through your working and round only the final answer.

Benefits and Limitations

Benefits

• Checks homework and textbook answers instantly, with steps you can follow.
• Simplifies radicals to exact form — the form teachers usually require.
• Handles the tricky cases: negative cube roots, higher nth roots, non-perfect radicands.
• Verifies mental-math traps like √144 + ∛27 before you answer them.
• Responsive on phones, tablets and desktops.

Limitations

• Works with real numbers only — even roots of negatives return "not a real number."
• Simplify Radicals mode handles whole-number radicands, not variable expressions.
• Decimal approximations are rounded to four places.

Quick Definitions

Related Math Tools

Square Root Calculator Methodology

This calculator evaluates roots with the standard mathematical definitions, then reduces results to exact simplified form where possible.

The radicand is prime-factorized; groups of n identical prime factors are moved outside the nth root (√50 = √(2 × 5²) = 5√2; ∛54 = ∛(2 × 3³) = 3∛2). Decimal approximations use double-precision arithmetic, rounded to four decimal places (√2 ≈ 1.41421, 5√2 ≈ 7.0711). The page is intended as an educational calculator and verification tool.

Mathematical References

• Khan Academy – Roots (8th grade math)
• Paul's Online Math Notes (Lamar University) – Radicals
• Math is Fun – Square Roots
• Math is Fun – Cube Roots