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.
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 $X$.
Since most calculators work with exponents, the practical formula is:
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.