Square Root Calculator
Find square roots, cube roots, and nth roots. Simplify radicals, check for perfect squares, and see if results are rational or irrational.
About Square Root Calculator
Find the square root, cube root, or nth root of any number and see the simplified radical form. The calculator factors out perfect squares, tells you whether the result is rational or irrational, and shows the decimal value to full precision.
How to Simplify a Square Root
Simplifying a radical means pulling perfect square factors out from under the root sign. The process works like this:
- Factor the number into a product that includes the largest perfect square
- Take the square root of the perfect square factor
- Write it outside the radical, leaving the remainder inside
Worked example: Simplify √72
- Find the largest perfect square factor of 72: 36 × 2 = 72
- √72 = √(36 × 2) = √36 × √2
- √36 = 6, so √72 = 6√2
- As a decimal: 6 × 1.41421... = 8.48528...
More examples:
| √n | Factorisation | Simplified Form | Decimal |
|---|---|---|---|
| √50 | 25 × 2 | 5√2 | 7.07107 |
| √48 | 16 × 3 | 4√3 | 6.92820 |
| √200 | 100 × 2 | 10√2 | 14.14214 |
| √98 | 49 × 2 | 7√2 | 9.89949 |
| √75 | 25 × 3 | 5√3 | 8.66025 |
| √180 | 36 × 5 | 6√5 | 13.41641 |
| √12 | 4 × 3 | 2√3 | 3.46410 |
Perfect Squares Reference
A perfect square has an integer square root. Memorising the first 20 is useful for quick mental maths and simplifying radicals:
| n | n² | n | n² | n | n² | n | n² |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 6 | 36 | 11 | 121 | 16 | 256 |
| 2 | 4 | 7 | 49 | 12 | 144 | 17 | 289 |
| 3 | 9 | 8 | 64 | 13 | 169 | 18 | 324 |
| 4 | 16 | 9 | 81 | 14 | 196 | 19 | 361 |
| 5 | 25 | 10 | 100 | 15 | 225 | 20 | 400 |
Rational vs Irrational Roots
A root is rational if it can be written as a fraction (whole numbers and terminating decimals). It is irrational if the decimal goes on forever without repeating.
| Type | Example | Why |
|---|---|---|
| Rational | √25 = 5 | 25 is a perfect square |
| Rational | √(9/16) = 3/4 | Both numerator and denominator are perfect squares |
| Irrational | √2 = 1.41421356... | 2 is not a perfect square |
| Irrational | √3 = 1.73205080... | 3 is not a perfect square |
| Irrational | √7 = 2.64575131... | 7 is not a perfect square |
The proof that √2 is irrational was one of the earliest results in mathematics, attributed to Hippasus around 500 BC. In fact, the square root of any non-perfect-square integer is irrational.
Cube Roots and How They Work
A cube root asks: what number multiplied by itself three times gives this value?
∛x = y means y × y × y = x
Worked example: Simplify ∛54
- Find the largest perfect cube factor: 27 × 2 = 54
- ∛54 = ∛(27 × 2) = ∛27 × ∛2
- ∛27 = 3, so ∛54 = 3∛2
- As a decimal: 3 × 1.25992... = 3.77976...
Perfect cubes to know:
| n | n³ | n | n³ |
|---|---|---|---|
| 1 | 1 | 6 | 216 |
| 2 | 8 | 7 | 343 |
| 3 | 27 | 8 | 512 |
| 4 | 64 | 9 | 729 |
| 5 | 125 | 10 | 1,000 |
Unlike square roots, cube roots of negative numbers are real. ∛(-8) = -2 because (-2) × (-2) × (-2) = -8.
Nth Roots
The nth root generalises the concept: the nth root of x is the number that, raised to the power n, gives x.
| Root | Notation | As Exponent | Example |
|---|---|---|---|
| Square root | √x | x^(1/2) | √16 = 4 |
| Cube root | ∛x | x^(1/3) | ∛27 = 3 |
| Fourth root | ⁴√x | x^(1/4) | ⁴√81 = 3 |
| Fifth root | ⁵√x | x^(1/5) | ⁵√32 = 2 |
| Sixth root | ⁶√x | x^(1/6) | ⁶√64 = 2 |
Even roots (2nd, 4th, 6th) of negative numbers are not real. Odd roots (3rd, 5th, 7th) of negative numbers are real and negative.
Where Square Roots Appear
| Application | Formula | Why Roots Are Needed |
|---|---|---|
| Pythagorean theorem | c = √(a² + b²) | Finding the hypotenuse or side of a right triangle |
| Distance formula | d = √((x₂-x₁)² + (y₂-y₁)²) | Distance between two points on a plane |
| Quadratic formula | x = (-b ± √(b²-4ac)) / 2a | The discriminant is under a square root |
| Standard deviation | σ = √(variance) | Variance is in squared units, root converts back |
| RMS voltage | V_rms = V_peak / √2 | Root mean square for AC circuits |
| Escape velocity | v = √(2GM/r) | Speed needed to leave a gravitational field |
Roots and the Exponent Connection
Every root can be written as a fractional exponent. This is the key to understanding roots in algebra:
ⁿ√x = x^(1/n)
This means all exponent rules apply to roots. For example, √a × √b = √(ab), and √(a/b) = √a / √b. The exponent calculator handles any base raised to any power, including fractional exponents. For problems involving right triangles, the Pythagorean theorem calculator uses square roots to find missing sides.
All calculations run entirely in your browser. No data is sent to any server.
Frequently Asked Questions
How does the radical simplification work?
The calculator factors out perfect squares from under the radical. For example, sqrt(72) becomes 6*sqrt(2) because 72 = 36 x 2 and sqrt(36) = 6.
Can I calculate cube roots and higher?
Yes. Use the Cube Root tab for cube roots, or the Nth Root tab for any root index like 4th root, 5th root, etc.
What does rational vs irrational mean?
A rational root is a whole number or fraction (like sqrt(25) = 5). An irrational root cannot be expressed as a fraction and has infinite non-repeating decimals (like sqrt(2) = 1.41421...).
Can I find the square root of negative numbers?
Square roots of negative numbers are not real numbers. They produce complex (imaginary) numbers. This calculator works with real numbers only.
What is a perfect square?
A perfect square is a number that has an integer square root. Examples are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. The reference table shows perfect squares from 1 to 20.
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