Square Root Calculator — Calculate √x and nth Roots
Calculate the square root, cube root, or any nth root of a number. Also shows whether the number is a perfect square or cube. See also Exponent Calculator and Log Calculator.
2 = square root, 3 = cube root, etc.
How to Calculate Square Roots
The square root of a number x is the value that, when multiplied by itself, gives x. For example, √144 = 12 because 12 × 12 = 144. This calculator also handles nth roots: the cube root (3rd root), 4th root, and beyond. Enter your number and the root degree, then click Calculate. The calculator will also tell you whether the number is a perfect square, cube, or nth power.
Square Root Formula
√x = x^(1/2)
∛x = x^(1/3)
ⁿ√x = x^(1/n)
Properties:
√(a × b) = √a × √b
√(a / b) = √a / √b
(√x)² = x
Example
√144 = ?
We need a number that multiplied by itself gives 144.
12 × 12 = 144 ✓
√144 = 12 (perfect square)
Frequently Asked Questions
What is a perfect square?
A perfect square is an integer that is the square of another integer. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. The square root of a perfect square is always a whole number.
Can you take the square root of a negative number?
Not in real numbers. The square root of a negative number is an imaginary number. For example, √(-1) = i (the imaginary unit). However, odd roots of negative numbers are real: ∛(-8) = -2.
What is the difference between square root and cube root?
A square root finds a number that multiplied by itself gives the original (2nd root). A cube root finds a number that multiplied by itself three times gives the original (3rd root). For example, √9 = 3 and ∛27 = 3.
Solved Examples — Square Roots
Example: Simplify √72
Solution:
Step 1: Find the largest perfect square factor of 72
Step 2: 72 = 36 × 2
Step 3: √72 = √(36 × 2) = √36 × √2 = 6√2
Step 4: As decimal: 6 × 1.4142 ≈ 8.485
Answer: 6√2 ≈ 8.485
Example: A square room has an area of 225 sq ft. What is the length of each side?
Solution:
Step 1: Area of square = side²
Step 2: side = √(Area) = √225
Step 3: 15 × 15 = 225 ✓
Answer: Each side is 15 feet
Example: Calculate ∛(216) (cube root of 216)
Solution:
Step 1: We need n such that n³ = 216
Step 2: Try 6: 6³ = 6 × 6 × 6 = 216 ✓
Answer: ∛216 = 6
Example: Find the distance between points (1, 2) and (4, 6) using the distance formula.
Solution:
Step 1: d = √((x₂−x₁)² + (y₂−y₁)²)
Step 2: d = √((4−1)² + (6−2)²) = √(9 + 16) = √25
Step 3: √25 = 5
Answer: Distance = 5 units
Practice Questions
Try these on your own:
- Find √196 (Answer: 14)
- Simplify √50 in simplified radical form (Answer: 5√2)
- Calculate ∛(1000) (Answer: 10)
- A circular garden has area 314 sq meters. What is the radius? Use π ≈ 3.14 (Answer: 10 m)
- Estimate √85 to one decimal place (Answer: 9.2)
- Simplify √(48/75) (Answer: 4/5)
Common Mistakes to Avoid
The most common mistake is thinking √(a + b) = √a + √b — this is incorrect! For example, √(9 + 16) = √25 = 5, not √9 + √16 = 3 + 4 = 7. Square roots do NOT distribute over addition or subtraction. However, they DO distribute over multiplication: √(a × b) = √a × √b. Another error is forgetting the ± when solving equations — if x² = 49, then x = ±7, not just 7. Students also confuse squaring and square rooting: (√5)² = 5, but √(5²) = |5| = 5 (the absolute value matters for negative inputs). When simplifying radicals, always look for the LARGEST perfect square factor, not just any factor — √72 is better written as 6√2 (using 36) than 2√18 (using 4).
Key Takeaways
- √n finds the number that, when multiplied by itself, gives n. It is the inverse of squaring.
- Perfect squares (1, 4, 9, 16, 25, 36, ...) have whole-number square roots.
- Product property: √(a × b) = √a × √b. Use this to simplify radicals.
- Square roots do NOT distribute over addition: √(a+b) ≠ √a + √b.
- The nth root of x can be written as x^(1/n). For example, ∛x = x^(1/3).
- Real-world applications include distance formula, area calculations, standard deviation, and the Pythagorean theorem.