Root Calculator

A root is the inverse operation of raising a number to a power. It asks: What number, when multiplied by itself 'n' times, equals our starting number? For example, the square root (degree 2) of 25 is 5 because 5 multiplied by itself (5 × 5) equals 25.

Yes, but only if the degree of the root (the index) is an odd number. For example, the cube root of -8 is -2, because -2 × -2 × -2 = -8. If the root index is even (like a square root or fourth root), the result requires complex or imaginary numbers, which are undefined in standard real-number mathematics.

To simplify a root like √18, look for the largest perfect square factor of the radicand. Since 9 is a perfect square and divides 18, we write √18 = √(9 × 2). Taking the square root of 9 gives us 3, leaving 2 under the radical: 3√2. Higher degree roots work similarly by searching for perfect cube, 4th, or nth power factors.

A rational root evaluates to a clean integer or simple fraction (like √16 = 4). An irrational root has a decimal expansion that goes on forever without repeating (like √2 ≈ 1.41421). Our calculator displays both the exact decimal approximation and the simplified radical structure.

Taking a root is mathematically identical to raising a base to a fractional power. The degree of the root becomes the denominator of the exponent. So, the square root of x is x raised to the power of 1/2 (x^0.5), and the cube root of x is x raised to the power of 1/3.