Free Math Tool
Square Root Calculator
Instantly find square roots, cube roots, and nth roots with step-by-step solutions, exact formulas, and perfect square detection.
√ Square Root
∛ Cube Root
ⁿ√ Nth Root
Step-by-Step
Perfect Square Checker
Irrational Numbers
√
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Step-by-Step Solution
∛
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Step-by-Step Solution
Root (n)
Number
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Step-by-Step Solution
Root Formulas & Identities
Mathematical definitions, rules, and relationships for root operations.
Square Root
√a = a^(1/2)
Value b where b² = a. Principal root is always non-negative. Example: √49 = 7, since 7² = 49.
Cube Root
∛a = a^(1/3)
Value b where b³ = a. Defined for all real numbers including negatives. Example: ∛(−8) = −2.
Nth Root
ⁿ√a = a^(1/n)
Generalized form. For odd n, real results exist for negative a. For even n, negative a yields imaginary results.
Product Rule
√(a · b) = √a · √b
Square root distributes over multiplication. Used to simplify radicals. Example: √(4·9) = 2·3 = 6.
Quotient Rule
√(a/b) = √a / √b
Square root distributes over division (b ≠ 0). Example: √(16/4) = 4/2 = 2.
Imaginary Root
√(−a) = i√a
Where a > 0 and i = √(−1). Imaginary numbers are the foundation of complex number mathematics.
How to Calculate by Hand
Manual methods using the Babylonian (Heron's) algorithm — converges rapidly to any precision.
Babylonian Method for Square Root
Heron's Algorithm
1
Choose an initial estimate
b near the expected answer.2
Divide the number by your estimate:
c = a ÷ b3
Average them:
b = (b + c) / 24
Repeat steps 2–3 until the value stabilises to the desired precision.
Worked Example: √27 to 4 decimal places
Estimate b = 5.125
27 ÷ 5.125 = 5.268 → avg = (5.125 + 5.268)/2 = 5.1963
27 ÷ 5.1963 = 5.196 → avg = (5.1963 + 5.196)/2 = 5.1962
27 ÷ 5.1962 = 5.1962 → √27 ≈ 5.1962 ✓
27 ÷ 5.125 = 5.268 → avg = (5.125 + 5.268)/2 = 5.1963
27 ÷ 5.1963 = 5.196 → avg = (5.1963 + 5.196)/2 = 5.1962
27 ÷ 5.1962 = 5.1962 → √27 ≈ 5.1962 ✓
Newton's Method for Nth Root
General Root
1
Start with an initial estimate
b.2
Compute:
c = a ÷ b^(n−1)3
Update:
b = [(n−1)×b + c] / n4
Repeat until the answer converges to the desired accuracy.
Worked Example: ⁵√100 to 3 decimal places
Estimate b = 2.5
100 ÷ 2.5⁴ = 2.56 → update = (4×2.5 + 2.56)/5 = 2.512
100 ÷ 2.512⁴ = 2.512 → converges → ⁵√100 ≈ 2.512 ✓
100 ÷ 2.5⁴ = 2.56 → update = (4×2.5 + 2.56)/5 = 2.512
100 ÷ 2.512⁴ = 2.512 → converges → ⁵√100 ≈ 2.512 ✓
Perfect Squares & Cubes Reference
Common values for n², √n, n³, and ∛n at a glance.
| n | n² (square) | √n (square root) | n³ (cube) | ∛n (cube root) |
|---|
Frequently Asked Questions
Common questions about square roots, cube roots, and root calculations.
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