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Home » Math & Conversions » Algebra » Square Root Square Root and Nth Root Calculator Find the square root ($\sqrt{x}$) or any arbitrary N-th root ($\sqrt[N]{x}$) of a number. Click 'Calculate' to see the exact decimal result and the simplified radical form. N-th Root (N): Radicand (X): Calculate Square Root ($\sqrt{X}$) - Nth Root ($\sqrt[N]{X}$) - Understanding Square Roots and Nth Roots The square root ($\sqrt{x}$) is the most common radical function. The **Nth Root** ($\sqrt[N]{x}$) is the generalized function that includes the square root (where $N=2$) and the cube root (where $N=3$). The Nth Root Formula The calculation is essentially finding a number $R$ that, when multiplied by itself $N$ times, equals the original number $X$. $$ R = \sqrt[N]{X} \quad \text{which is equivalent to} \quad R^N = X $$ Since most calculators work with exponents, the practical formula is: $$ \sqrt[N]{X} = X^{\frac{1}{N}} $$ Simplifying Radicals (Square Roots) When solving algebra problems, you often need the **simplified radical form** (e.g., $5\sqrt{2}$) rather than the decimal answer (e.g., 7.071). To simplify a square root, you find the largest perfect square factor of the radicand. Example: Simplify $\sqrt{50}$ Identify factors of 50: (1, 50), (2, 25), (5, 10). Find the largest perfect square factor: 25. Rewrite the radical: $\sqrt{50} = \sqrt{25 \cdot 2}$. Take the square root of the perfect square: $\sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$. Our calculator performs this simplification for you. Frequently Asked Questions (FAQ) What is the formula for square root? The formula is $\sqrt{X}$. It is calculated by raising the number $X$ to the power of $\frac{1}{2}$ ($X^{1/2}$). Is $\sqrt{2}$ an irrational number? Yes. $\sqrt{2}$ (approximately 1.41421...) is an **irrational number**. This means it cannot be expressed as a simple fraction (a/b) and its decimal expansion is non-terminating and non-repeating. Can you find the square root of a negative number? In real numbers, **no**. The square root of a negative number (e.g., $\sqrt{-4}$) is an **imaginary number** ($2i$). Our calculator returns an error for negative inputs. Related Calculators Roots Calculator (Nth Root) Exponent Calculator Logarithm Calculator Function Calculator Matrix Calculator Mode Calculator Range Calculator Factoring Calculator Decimal to Fraction
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