What is a perfect square?

A perfect square is an integer that is the square of another integer. Examples: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Their square roots are all whole numbers.

What does simplified radical form mean?

Simplified radical form means pulling out any perfect-square factors from under the radical. For example, √12 = √(4×3) = 2√3. The coefficient is 2 and the radicand is 3.

How do you calculate the cube root?

Set the root index (n) to 3. The cube root of 27 = 27^(1/3) = 3. The cube root of 8 = 2.

Can you take the square root of a negative number?

In the real number system, negative numbers don't have square roots. This calculator requires a non-negative input. Imaginary numbers (i = √(−1)) are a separate topic in complex number theory.

Square Root Calculator

A square root calculator finds the square root — or any nth root — of a number. It also checks whether the number is a perfect square and expresses the result in simplified radical form. Whether you need √144 for a geometry problem or ∛216 for a volume calculation, this tool gives you the exact answer instantly.

What Is a Square Root?

The square root of a number x is a value r such that r² = x. Every non-negative number has exactly one non-negative square root, denoted √x. Negative numbers have no real square roots.

The nth root (or n-th root) generalizes this concept: the nth root of x is a value r such that rⁿ = x, written ⁿ√x or x^(1/n).

Perfect Squares

A perfect square is a non-negative integer whose square root is also an integer. The first 15 perfect squares are:

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196

You can quickly check: is 225 a perfect square? √225 = 15 (integer) → yes. Is 50 a perfect square? √50 ≈ 7.071 (not integer) → no.

Simplified Radical Form

A radical expression is simplified when:

The radicand (number under the radical) has no perfect-square factor other than 1.
There are no fractions under the radical.

How to Simplify √n

Factor the radicand into primes, then pull out pairs (for √) or groups of n (for nth root).

Example: Simplify √72

72 = 2 × 36 = 2 × 6² → Wait, 72 = 4 × 18 = 4 × 9 × 2 = 36 × 2
√72 = √(36 × 2) = 6√2

Algorithm (for any integer x and nth root n):

Find prime factorization of x: x = p₁^e₁ × p₂^e₂ × …
For each prime pᵢ with exponent eᵢ:
- coefficient factor: pᵢ^⌊eᵢ/n⌋
- radicand factor: pᵢ^(eᵢ mod n)
Multiply coefficient factors together; multiply radicand factors together.

How to Calculate Square Roots by Hand

Method 1 — Guess and Check

For √50: Try 7² = 49, 8² = 64. So √50 is between 7 and 7.1. Try 7.07² = 49.98, 7.08² = 50.13 → √50 ≈ 7.07.

Method 2 — Babylonian (Newton’s) Method

Start with a guess g₀. Repeatedly apply: gₙ₊₁ = (gₙ + x/gₙ) / 2

Example: √2 starting with g₀ = 1.5

g₁ = (1.5 + 2/1.5) / 2 = (1.5 + 1.333) / 2 = 1.4167
g₂ = (1.4167 + 2/1.4167) / 2 ≈ 1.4142

This converges quadratically and is very fast.

Worked Examples

Example 1 — Architecture: Staircase Diagonal

A staircase rises 3 m and runs 4 m horizontally. What is the length of the railing?

By the Pythagorean theorem: diagonal = √(3² + 4²) = √(9 + 16) = √25 = 5 m

Example 2 — Simplifying √180

180 = 4 × 45 = 4 × 9 × 5 = 36 × 5
√180 = √(36 × 5) = 6√5
Coefficient: 6, Radicand: 5
Decimal: 6 × √5 ≈ 6 × 2.2361 ≈ 13.416

Example 3 — Cube Root of 343

ⁿ√343 with n = 3: 7³ = 343. So ∛343 = 7 (perfect cube).

Example 4 — nth Root in Finance

The formula for CAGR (Compound Annual Growth Rate) uses nth roots: CAGR = (Ending Value / Beginning Value)^(1/n) − 1

If an investment grew from $1000 to$ 1600 in 5 years: CAGR = (1600/1000)^(1/5) − 1 = 1.6^0.2 − 1 ≈ 0.0986 = 9.86% per year

Common Mistakes

√(a + b) ≠ √a + √b. For example, √(9 + 16) = √25 = 5, not 3 + 4 = 7.
√(a²) = |a|, not always a. If a is negative, √(a²) = −a, not a.
Forgetting to check for perfect square factors. Always prime-factorize before simplifying.
Confusing the index. √x (index 2) vs ∛x (index 3) — the index is not written for square roots by convention.

Frequently Asked Questions

Why can’t you take the square root of a negative number in real numbers? Because any real number squared is non-negative. There’s no real number r such that r² = −1. In complex number theory, i = √(−1) is defined, opening a whole new field of mathematics.

What is the difference between exact and approximate roots? Exact: √4 = 2 (rational, terminating). √2 = 1.41421356… is irrational — the decimal never repeats or terminates. For exact work, leave it as √2 (simplified radical form).

How accurate is the nth root calculation? This calculator uses JavaScript’s Math.sqrt (for n=2) and Math.pow(x, 1/n) for other roots, which are IEEE 754 double-precision operations accurate to about 15 significant digits. Results are displayed to 6 decimal places.

Can I compute fractional roots (e.g., x^(1/3))? Yes — set n=3 for cube root, n=4 for 4th root, etc. The calculator supports integer root indices from 2 to 100.