What is a perfect square?

A perfect square is an integer that can be expressed as the square of another integer. Examples include 1, 4, 9, 16, 25, 36, and 49.

Can you take the square root of a negative number?

In the real number system, you cannot because any real number multiplied by itself is positive. However, in the complex number system, negative square roots are defined using the imaginary unit i (where i² = -1), e.g., √(-9) = 3i.

What is the principal square root?

Every positive number has two square roots (one positive and one negative). The principal square root is specifically the positive square root, denoted by the radical symbol √.

Math Tool

Square Root Calculator

Find the square root of any number. Quickly calculate roots, simplifies radicals, identifies perfect squares, and provides step-by-step estimations.

Interactive Square Root Calculator Workspace

Live Solver

Enter Radicand (Value under root)

Principal Square Root√80 ≈ 8.944272

Simplified Radical form4√5

Is Perfect Square?No (Irrational)

Newton-Raphson Root Estimation Proof

To approximate √80, we use Newton\'s Method starting with a guess of y₀ = 9:

First iteration: y₁ = (y₀ + 80/y₀) / 2 = 8.944444
Actual Value: √80 = 8.9442719100

Mathematical Formula & Variables

Equation Model√x = y => y² = x

The square root of a number x is a number y such that when y is multiplied by itself, it equals x. Negative values return complex number roots.

Variable Definitions

Symbol	Description
x	The radicand value
y	The resulting principal square root

How to Use the Square Root Calculator

Enter the radicand value (positive or negative).
The calculator will immediately compute the positive principal root and its negative root.
It also determines if the number is a perfect square, simplifies the radical to its simplest coefficient form (e.g., √80 = 4√5), and details the Newton-Raphson approximation steps.

Practical Example Calculation

Scenario Context: Calculating the square root of 80 and simplifying its radical form.

Step 1: Input number: 80.

Step 2: Find the highest perfect square factor of 80, which is 16 (80 = 16 × 5).

Step 3: Apply product rule: √80 = √(16 × 5) = √16 × √5.

Step 4: Calculate perfect square root: √16 = 4.

Step 5: Combine results: 4√5.

The square root of 80 is approximately 8.9443, or 4√5 in simplified radical terms.

Frequently Asked Questions

Support CalculatorHub

Enjoying this calculator? Support our work to keep all 167+ calculation tools 100% free and open for everyone.

Support on Ko-fi

Related Calculators

Cube Root CalculatorQuick solver tool

Scientific CalculatorQuick solver tool

Exponent CalculatorQuick solver tool