Root Calculator
Calculate square root, cube root, or any nth root instantly — with formula and verification.
You might also need
How It Works
The square root of a number n is the value that, when multiplied by itself, equals n. Example: √144 = 12 because 12 × 12 = 144. Used to find side lengths of squares, solve quadratic equations, and calculate distances.
√n = n^(1/2)√144 = 12 | √2 ≈ 1.41421 | √50 ≈ 7.07107The cube root of n is the value that, when multiplied by itself three times, equals n. Example: ∛27 = 3 because 3 × 3 × 3 = 27. Used to find the side length of a cube from its volume.
∛n = n^(1/3)∛27 = 3 | ∛8 = 2 | ∛125 = 5The nth root is the generalized form. Enter any root value. Example: the 4th root of 81 = 3, because 3^4 = 81. Finance use: the 10th root of 2.0 tells you the annual growth rate needed to double in 10 years (≈ 7.18%/year).
ⁿ√x = x^(1/n)4√81 = 3 | 5√32 = 2 | 10√1024 ≈ 2Quick Reference
Common root calculations — results you can verify instantly.
Perfect square
√4
2
Perfect square
√9
3
Perfect square
√25
5
Classic example
√144
12
Perfect cube
∛8
2
Perfect cube
∛27
3
Perfect cube
∛125
5
4th root
4√81
3
Tips & Shortcuts
For perfect squares (4, 9, 16, 25, 36...), the square root is always a whole number. Memorize up to √144 = 12 to speed up mental math.
To estimate √n mentally: find the nearest perfect square below and above, then interpolate. √50 is between √49=7 and √64=8, so ≈ 7.07.
Cube roots of perfect cubes (8, 27, 64, 125) are whole numbers. ∛8=2, ∛27=3, ∛64=4, ∛125=5 — worth memorizing.
The nth root formula ⁿ√x = x^(1/n) works for any root. To find CAGR: use nth root where n = number of years.
Square root and squaring are inverse operations. If √x = y, then y² = x. Always verify your answer by squaring it back.
Common Mistakes
Confusing square root with dividing by 2
√100 = 10, not 50. Square root asks: what number times itself equals 100? It is the inverse of squaring, not division.
Expecting √(a+b) = √a + √b
This is wrong. √(9+16) = √25 = 5, not √9 + √16 = 3 + 4 = 7. Roots do not distribute over addition.
Trying to find the square root of a negative number
√(−9) has no real solution. Negative numbers have imaginary square roots. Switch to complex number mode if needed.
Rounding the root before using it in further calculation
If √2 ≈ 1.414 and you use 1.4 in the next step, errors compound. Always use full precision until the final answer.
Confusing the cube root with cubing
∛27 = 3 (inverse of cubing). 27^3 = 19,683 (cubing). These are opposite operations.
Frequently Asked Questions
The square root of 144 is 12. Because 12 × 12 = 144. Formula: √144 = 144^(1/2) = 12. Other common examples: √25 = 5, √36 = 6, √64 = 8, √100 = 10.
The cube root of 27 is 3. Because 3 × 3 × 3 = 27. Formula: ∛27 = 27^(1/3) = 3. Other examples: ∛8 = 2, ∛64 = 4, ∛125 = 5.
Enter the number and the root (n). Formula: ⁿ√x = x^(1/n). Example: the 4th root of 81 = 81^(1/4) = 3. Verification: 3^4 = 81 ✓. The calculator includes automatic verification.
Not in real numbers. The square root of a negative number is imaginary (e.g., √−1 = i). This calculator works with non-negative real numbers only. For complex roots, a scientific calculator is required.
Square root (√) finds the number that, multiplied by itself, gives the original. Cube root (∛) finds the number multiplied by itself three times. Both are special cases of the nth root formula: ⁿ√x = x^(1/n).
For perfect squares, memorize common values: √1=1, √4=2, √9=3, √16=4, √25=5, √36=6, √49=7, √64=8, √81=9, √100=10. For non-perfect squares, use the long division method or Newton's approximation method.
The square root of 2 is approximately 1.41421356. It is an irrational number — its decimal expansion never ends or repeats. It appears frequently in geometry, especially in diagonal calculations for squares.
Related Calculators
Midpoint Calculator
Find the midpoint between two coordinate points.
Log Calculator (base 10)
Calculate log base 10 for any positive number.
Scientific Calculator
Full scientific calculator with trig, log, and advanced functions.
Power Calculator
Calculate electrical power in watts from volts and amps.
Voltage Divider Calculator
Calculate output voltage of a resistor voltage divider.
Basic Calculator
Fast arithmetic: add, subtract, multiply, divide.