Root Calculator Online Free Tool
Root Calculator
Root Calculations
Square Root Calculator
The root calculator finds the nth root of any number. Square roots (n=2) and cube roots (n=3) are the most common, but this tool handles any root from the 2nd through the 10th and beyond. It also simplifies radical expressions and converts between root notation and fractional exponent form. Roots appear throughout algebra, geometry (diagonal lengths, circle areas), physics, statistics (standard deviation), and finance (CAGR calculations).
How Roots Work
The nth root of a number x is the value that, when raised to the nth power, gives x. Roots are the inverse operation of exponentiation. Every positive number has exactly one positive real nth root. Negative numbers have real nth roots only when n is odd: the cube root of -8 is -2, but the square root of -4 has no real value.
ⁿ√x = x^(1/n) Square root: √x = x^(1/2) Cube root: ³√x = x^(1/3) Fourth root: ⁴√x = x^(1/4) Simplification: √72 = √(36×2) = 6√2
An even root of a negative number is not real. Odd roots of negative numbers are real and negative: ³√-8 = -2.
Perfect Squares and Cubes Reference
Knowing perfect squares up to 20 and perfect cubes up to 10 by memory speeds up mental math and radical simplification significantly. A number is a perfect square if its square root is a whole number, and a perfect cube if its cube root is a whole number.
| n | n² (perfect square) | n³ (perfect cube) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 5 | 25 | 125 |
| 6 | 36 | 216 |
| 7 | 49 | 343 |
| 8 | 64 | 512 |
| 9 | 81 | 729 |
| 10 | 100 | 1,000 |
| 12 | 144 | 1,728 |
| 15 | 225 | 3,375 |
| 20 | 400 | 8,000 |
Simplifying Radical Expressions
A radical expression is simplified when no perfect square (or perfect cube for cube roots) remains under the radical sign. The process is to factor the radicand into the largest perfect square times a remainder, then take the square root of the perfect square factor out in front. Simplifying radicals is essential before adding or subtracting them: you can only combine like radicals (same index and same radicand).
Steps to simplify √180: 180 = 36 × 5 (36 is the largest perfect square factor) √180 = √(36 × 5) = √36 × √5 = 6√5 Adding radicals: 3√5 + 2√5 = 5√5 Different radicals cannot be combined: √5 + √3 ≠ √8
Always find the LARGEST perfect square factor to simplify in one step rather than multiple steps.
Roots in Real-World Applications
Square roots appear in geometry when calculating diagonal distances (Pythagorean theorem: c = √(a² + b²)), in physics for wave and energy calculations, and in finance for the Compound Annual Growth Rate formula. Cube roots are used in volume problems: finding the side length of a cube from its volume. The nth root appears in CAGR calculations: end value divided by start value, raised to the power of 1/n years.
Pythagorean theorem: c = √(a² + b²) Cube side from volume: s = ³√V CAGR = (End Value / Start Value)^(1/n) - 1
Frequently Asked Questions
What is the difference between a square root and a cube root?⌄
The square root (√x) finds the value that when multiplied by itself gives x. The cube root (³√x) finds the value that when multiplied by itself three times gives x. √9 = 3 because 3 × 3 = 9. ³√27 = 3 because 3 × 3 × 3 = 27. The key distinction is the index: square roots have index 2 (usually written without showing the 2), cube roots have index 3. Any positive number has exactly one positive real square root and one positive real cube root.
Why does the square root of a negative number not exist in real numbers?⌄
No real number multiplied by itself produces a negative result. Positive × positive = positive. Negative × negative = also positive. So there is no real number whose square is negative. The square root of -1 is defined as the imaginary unit i in complex number mathematics. Complex numbers of the form a + bi extend the real number line to a 2D plane and are used in electrical engineering, quantum mechanics, and signal processing.
How do I simplify a square root?⌄
Find the largest perfect square that divides evenly into the number under the radical. Take the square root of that perfect square factor and move it out in front. Leave what remains under the radical. Example: √72. Factors of 72: 4×18, 9×8, 36×2. The largest perfect square factor is 36. √72 = √(36×2) = √36 × √2 = 6√2. Always aim for the largest perfect square in one step rather than simplifying in multiple smaller steps.
What is a rational vs irrational root?⌄
A root is rational if the result can be expressed as a fraction or whole number. √9 = 3 (rational), √(1/4) = 1/2 (rational). A root is irrational if the result is a non-repeating, non-terminating decimal. √2 ≈ 1.41421356... (irrational). Any square root of a number that is not a perfect square is irrational. The square roots of 2, 3, 5, 6, 7, 8, and all other non-perfect-square integers are irrational numbers.
How do I estimate a square root without a calculator?⌄
Find the two perfect squares your number falls between. For √50: 49 = 7² and 64 = 8², so √50 is between 7 and 8, closer to 7. Linear interpolation: √50 ≈ 7 + (50-49)/(64-49) = 7 + 1/15 ≈ 7.07. The actual value is 7.071. For a quick first estimate, use the nearest perfect square and know the result is within about 1 unit of that square root.