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Square Root Calculator

Compute square roots, cube roots, and nth roots with principal value, simplified radical form, and imaginary handling.

MATH

Calculate the square root, cube root, or any nth root (up to n = 10) of a real number. Returns the principal root, the simplified radical form using the largest perfect-nth-power factor, and a decimal approximation. Even roots of positive numbers show both the positive and negative real solutions; even roots of negative numbers are handled as imaginary numbers using i = √-1.

The nth root of x is the value y such that y^n = x. For even n with positive x there are two real solutions (±y); the principal root is the non-negative one (this is what the radical symbol denotes). For odd n the result inherits the sign of x. Simplified radical form pulls the largest perfect-nth-power factor out from under the radical: for √72, the largest perfect square dividing 72 is 36, so √72 = √(36 × 2) = 6√2. Negative inputs to even roots produce imaginary numbers of the form bi. Worked example: √72 = 6√2 ≈ 8.485; ∛125 = 5; √-9 = 3i.

Disclaimer: Educational math utility. Floating-point arithmetic introduces tiny rounding errors for very large numbers or high root indices - the simplified radical form is the exact value.

Square Root Calculator

Compute the square root, cube root, or any nth root of a number. Get the principal value, simplified radical form, and handle negative inputs with imaginary numbers when needed.

Number (x)Any real number. Negative values allowed.

Root Index (n)2 = square root, 3 = cube root, up to 10. Must be an integer >= 2.

Quick examples:

Principal Root

8.485281

Simplified Radical Form6√2

Decimal Approximation8.485281

Both Real Roots (even n, positive x)± 8.485281

Roots, Radicals, and Imaginary Numbers

The nth root of x is the value y such that y^n = x. For even roots (square root, fourth root, etc.) of a positive number, there are actually two real solutions - one positive and one negative - because both (+y)^n and (-y)^n equal x. By convention the "principal root" is the non-negative one, which is what the radical symbol √ denotes. So √9 = 3 (principal), but the equation y^2 = 9 has two solutions: y = 3 or y = -3, often written as ±3. For odd roots there is only one real value, and it carries the sign of the input: ∛-8 = -2.

Simplified radical form rewrites a root by pulling out the largest perfect nth-power factor. For √72, factor 72 = 36 × 2 where 36 = 6^2 is the largest perfect square divisor. Then √72 = √(36 × 2) = √36 × √2 = 6√2. This is the exact value, while the decimal 8.4852... is only an approximation. Math teachers, the SAT, and proof-based courses generally want the simplified radical because it is exact and reveals structure. For nth roots, you look for the largest factor that is a perfect nth power: for ∛54, factor 54 = 27 × 2 = 3^3 × 2, so ∛54 = 3∛2.

Negative numbers under an even root have no real solution because any real number squared (or to any even power) is non-negative. Mathematicians extend the real numbers by defining the imaginary unit i = √-1, with the rule i^2 = -1. Then √-9 = √(9 × -1) = 3i, and √-72 = 6i√2. Numbers of the form a + bi are called complex numbers and form the foundation of electrical engineering (AC circuit analysis), quantum mechanics, signal processing (Fourier transforms), and large parts of pure mathematics. Negative inputs to odd roots stay real because an odd power of a negative number is still negative: ∛-27 = -3 with no imaginary part needed.

Educational math utility. Floating-point arithmetic introduces tiny rounding errors for very large numbers or high root indices - the simplified radical form is the exact value.

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Frequently Asked Questions

What is the principal square root?

For a positive number x, the equation y^2 = x has two solutions, one positive and one negative. The principal square root is the non-negative one - that is what the √ symbol denotes. So √25 = 5, even though y^2 = 25 has solutions y = ±5. The same rule extends to any even nth root: the principal value is the non-negative one.

What is a simplified radical form?

A radical is in simplified form when no perfect nth-power factor remains under the root symbol. For square roots, that means no perfect-square factor other than 1. To simplify √72, find the largest perfect square dividing 72 (which is 36): √72 = √(36 × 2) = 6√2. Simplified radicals are exact, unlike decimal approximations, and are the standard form expected in algebra and on standardized tests like the SAT.

Can I take the square root of a negative number?

Not in the real numbers - any real number squared is non-negative, so no real y satisfies y^2 = -9. Mathematicians extend the real numbers by defining the imaginary unit i = √-1, with i^2 = -1. Then √-9 = √9 × √-1 = 3i. Numbers of the form a + bi are called complex numbers and are essential in electrical engineering, quantum mechanics, and signal processing. Odd roots of negative numbers stay real: ∛-27 = -3.

What's the difference between square root and cube root?

A square root reverses squaring (raising to the 2nd power); a cube root reverses cubing (raising to the 3rd power). Because the index is different, behavior on negative numbers differs: square root (even index) of a negative is imaginary, but cube root (odd index) of a negative is just a negative real number, since (-x)^3 = -x^3. Also, square roots of positive numbers have two real solutions (±), while cube roots have exactly one real solution.

How do I compute a square root by hand?

The classical method is the digit-by-digit (long-division) algorithm, but in practice most students use one of two shortcuts: (1) factor the number and pull out perfect squares to get the simplified radical (√72 → 6√2), then leave it exact; or (2) use Newton's method - start with an estimate y, then iterate y := (y + x/y) / 2. For x = 72 starting at y = 8: next estimate is (8 + 9)/2 = 8.5, then (8.5 + 8.4706)/2 ≈ 8.4853, converging to √72 quickly.

Square Root Calculator

Square Root Calculator

Roots, Radicals, and Imaginary Numbers

Frequently Asked Questions

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