Math & ArithmeticVerified: 2026-10-05
Square Root & Exponent Calculator
Calculate principal square roots, cube roots, n-th radicals, simplified radical forms (k√r), powers (xʸ), and scientific notations.
Direct Answer:The square root of a number x is a value y such that y × y = x (√x = y). For example, √144 = 12. For non-perfect squares like √72, the simplified radical is 6√2 ≈ 8.4853.
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Adjust Parameters
Enter positive number to take root of
220
2 = Square Root (√), 3 = Cube Root (∛)
√144 Square Root
12
Perfect Root: 12² = 144
Simplified Radical Form
12
Perfect Root?
Yes (Integer Root)
Root Degree (n)
2
Step-by-Step Mathematical Work
- Expression: √144
- 144 is a perfect square: 12² = 144
- Principal square root: √144 = 12
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How Square Root & Exponent Calculator is Calculated
Transparent mathematical formulas and assumptions used across our financial engines.
Mathematical Formula
\sqrt[n]{x} = x^{1/n} \quad | \quad x^y = x \times x \times \dots \times x \quad | \quad \sqrt{a^2 b} = a \sqrt{b}
Variables & Legend
x
Radicand / Base
n
Root Degree
y
Exponent
a\sqrt{b}
Simplified Radical
Step-by-Step Calculation
- Factor Perfect Squares:
x = a^2 \times bFind the largest perfect square factor dividing the radicand.
- Extract Outer Factor:
\sqrt{x} = a\sqrt{b}Extract the square root of the square factor outside the radical.
- Compute Decimal:
x^{1/n}Evaluate principal real root as decimal.
Conventions & Assumptions
- In the real number system, the square root of a negative number is undefined.
- Every positive number has two square roots (±√x); this calculator computes the principal (positive) square root.
Frequently Asked Questions
Find the largest perfect square factor (4, 9, 16, 25, 36, etc.) that divides the number evenly. Take the square root of that factor outside the radical. For example, √72 = √(36 × 2) = 6√2.