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Square Root Calculator

Calculate square and nth roots

About the Calculator

Roots show up in algebra, geometry, and real world formulas, but the math can be slow without a calculator. This tool finds square roots and any nth root quickly, so you can focus on solving the bigger problem. Use it for homework, engineering formulas, or quick checks in class or at work. It also helps you see the difference between perfect squares and irrational results. Enter a number, set the root you want, and get an immediate answer. It is a simple way to keep your work accurate without breaking your flow. Use the Square Root Calculator to get a clear result you can act on right away.

Square Root

Square Root

12

Verification: 12^2 = 144

The Formula

Use these power forms on a calculator, or the manual methods below when you want to work roots out on paper.
√n = n^(1/2) | ∛n = n^(1/3) | ⁿ√x = x^(1/n)

Finding roots by hand

A calculator uses the same math you can do on paper: locate the root between two known powers, simplify factors when possible, or repeat a short iteration for square roots. Below are the methods used in the worked examples on this page.

1. Perfect powers (exact answers)

If the number equals an integer raised to the root index, the answer is that integer.

  • Square roots: test 1², 2², 3²… until you hit the target (e.g. 12² = 144, so √144 = 12).
  • Cube roots: test 1³, 2³, 3³… (e.g. 3³ = 27, so ∛27 = 3).
  • Always verify by raising your answer back to the root power.

2. Simplify with factors (still exact)

Split the number into a perfect power times a leftover factor, then use ⁿ√(a × b) = ⁿ√a × ⁿ√b.

50 = 25 × 2, and 25 = 5²
√50 = √25 × √2 = 5√2

18 = 9 × 2, and 9 = 3²
√18 = √9 × √2 = 3√2

Prime numbers like 13 have no perfect-square factor other than 1, so √13 cannot be shortened further; it stays √13 in radical form.

3. Bracket between two integers

When there is no exact root, trap the value between two consecutive powers.

  • √13: 3² = 9 and 4² = 16, so √13 is between 3 and 4 (closer to 4).
  • ∛29: 3³ = 27 and 4³ = 64, so ∛29 is between 3 and 4 (just above 3).
  • ∛50: between 3³ = 27 and 4³ = 64, nearer to 4.

Then test decimals (3.6², 3.07³, etc.) or use a calculator for the last digits. Bracketing tells you the right whole-number part and how close you are.

4. Babylonian method (square roots only)

Also called Heron's method. Pick a starting guess x, then repeat until stable:

xnew = (x + n ÷ x) ÷ 2

Example: approximate √13

  1. Start with x = 3.5 (between 3 and 4).
  2. First pass: x = (3.5 + 13 ÷ 3.5) ÷ 2 = (3.5 + 3.714…) ÷ 2 ≈ 3.607.
  3. Second pass: x = (3.607 + 13 ÷ 3.607) ÷ 2 ≈ 3.6056 (already very close).
  4. Square to check: 3.6056² ≈ 13.0004.

Each repetition roughly doubles correct digits. Two or three rounds are enough for homework-level accuracy.

5. Cube roots and higher roots

There is no equally short pencil method like Babylonian for cube roots. In practice you bracket with perfect cubes (or n-th powers), make a decimal guess, and cube it to see if you are high or low. For homework, bracketing plus one or two trial values is usually enough; for precision, use the power rule n^(1/k) on a calculator, which is the same as the formula shown above.

Quick map: perfect power → exact integer; factorable number → simplify (like 5√2); otherwise → bracket, then Babylonian for squares or trial cubes for cube roots. The worked examples below walk through each path step by step.

Examples

What is √144?

  1. A square root asks: what number multiplied by itself equals 144? In symbols, find x where x² = 144.
  2. Test nearby perfect squares: 10² = 100 (too small), 11² = 121 (still small), 12² = 144.
  3. So the exact value is 12.
  4. Verify: 12 × 12 = 144, so the root checks out.

Answer: 12

What is ∛27?

  1. A cube root asks: what number cubed equals 27? Find x where x³ = 27.
  2. Try small integers: 1³ = 1, 2³ = 8, 3³ = 27.
  3. Therefore the cube root is 3.
  4. Verify: 3 × 3 × 3 = 27.

Answer: 3

What is √50?

  1. Start by factoring 50 into a perfect square times another number: 50 = 25 × 2.
  2. Use the product rule: √50 = √(25 × 2) = √25 × √2 = 5√2 (exact simplified form).
  3. For a decimal estimate, √2 ≈ 1.41421356, so 5√2 ≈ 5 × 1.41421356.
  4. Multiply: 5 × 1.41421356 ≈ 7.07106781, usually rounded to about 7.071.
  5. Check with the calculator rule: 50^(1/2) ≈ 7.07106781.

Answer: 5√2 (exact) ≈ 7.071 (decimal)

What is √13?

  1. 13 is prime, so it cannot be split into a larger perfect square times a smaller whole number. The exact form stays √13.
  2. Bracket the value with nearby squares: 3² = 9 and 4² = 16, so √13 is between 3 and 4 (closer to 4 because 13 is nearer to 16 than to 9).
  3. A quick linear estimate: 13 is about 44% of the way from 9 to 16, so try roughly 3 + 0.44 ≈ 3.44 as a first guess.
  4. Square the guess to check: 3.44² ≈ 11.83 (too low). Try 3.6² = 12.96 (still low). Try 3.61² ≈ 13.03 (very close).
  5. Refine with the power rule on a calculator: 13^(1/2) ≈ 3.605551275.

Answer: √13 (exact) ≈ 3.606 (decimal)

What is ∛29?

  1. 29 is not a perfect cube. Find x where x³ = 29 by bracketing with nearby cubes.
  2. 2³ = 8 (too small), 3³ = 27 (just below 29), 4³ = 64 (too large). So ∛29 is between 3 and 4, just above 3.
  3. Because 29 is only 2 more than 27, the root is only slightly greater than 3 (much closer to 3 than to 4).
  4. Test 3.1³ = 29.791 (a bit high) and 3.07³ ≈ 28.99 (very close to 29).
  5. Using the nth-root rule: 29^(1/3) ≈ 3.072316826.

Answer: ∛29 (exact) ≈ 3.072 (decimal)

What is √18?

  1. Factor 18: 18 = 9 × 2, and 9 is a perfect square.
  2. Apply the product rule: √18 = √(9 × 2) = √9 × √2 = 3√2.
  3. Because 9 is the largest perfect-square factor of 18, 3√2 is the simplest radical form.
  4. Decimal form: √2 ≈ 1.41421356, so 3√2 ≈ 3 × 1.41421356 ≈ 4.242640687.
  5. Verify: 4.242640687² ≈ 18.

Answer: 3√2 (exact) ≈ 4.243 (decimal)

What is ∛50?

  1. 50 is not a perfect cube. Use nearby cubes to locate the answer: 3³ = 27 and 4³ = 64.
  2. Since 50 lies between 27 and 64, ∛50 is between 3 and 4. It is nearer to 4 (64 − 50 = 14) than to 3 (50 − 27 = 23).
  3. Try 3.6³ = 46.656 (low) and 3.7³ ≈ 50.653 (slightly high), so the value is near 3.68.
  4. Calculator check: 50^(1/3) ≈ 3.684031499.
  5. Unlike square roots of composites like 50, cube roots of 50 do not simplify to a clean integer coefficient times a smaller root.

Answer: ∛50 (exact) ≈ 3.684 (decimal)

Tips & Strategies

Perfect squares to memorize: 1. 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.

√(a×b) = √a × √b. useful for simplifying.

Quick tip. Negative numbers don't have real square roots (but do have imaginary roots).

Quick tip. Cube roots of negative numbers are negative.

Cross-check when the decision matters. Run a second scenario with rounded inputs or a different path to the same quantity so you do not rely on a single fragile chain of arithmetic.

Things Worth Knowing

  • •√2 Broke Greek Mathematics: Around 500 BC, Pythagoreans discovered √2 = 1.41421356... is irrational (never-ending, non-repeating decimals), shattering their belief that all numbers were fractions and allegedly causing mathematician Hippasus to be drowned for revealing it.
  • •Pythagorean Theorem Everywhere: The formula a² + b² = c² (requiring square roots) is used millions of times daily in GPS navigation, construction, video game graphics, and even determining baseball diamond distances, making it perhaps the most-used math theorem in modern life.
  • •The Symbol's Typography Origin: The square root symbol √ was introduced in 1525 by mathematician Christoph Rudolff and comes from a stylized lowercase 'r' (for "radix" meaning root in Latin); it's essentially a 500-year-old abbreviation.
  • •Sprint Speed Square Root: Your maximum running speed is roughly proportional to the square root of your leg length; which is why horses (√100cm ≈ 10) run faster than humans (√90cm ≈ 9.5), despite having only 11% longer legs.
  • •Inverse Square Law: Light intensity decreases by the square of distance; so a lamp 2 feet away is ¼ as bright as at 1 foot (1/2² = 1/4), and at 3 feet it's 1/9 as bright (1/3² = 1/9), explaining why rooms get dark so quickly.
  • •Video Game Distance Formula: Every video game uses √(Δx² + Δy²) millions of times per second to calculate distances between characters, bullets, and objects; without square roots, 3D games wouldn't exist.