Square Root Calculator - Simplify Radicals & Find Roots

Square Root Calculator

Smart calculation with radical simplification, complex roots, and batch tables.

How to Use This Calculator

  1. Calculator Mode: Enter any number to find its root. Use the toggles to see negative roots or step-by-step logic.
  2. Batch Mode: Switch tabs to generate a reference table. Useful for teachers or students needing a list of squares/roots from 1 to 100.
  3. Toolbox Mode: Use this to simplify specific expressions like rationalizing the denominator (e.g., input 1/sqrt(3)).

Example of Using This Calculator: Simplifying the Square Root of 72

Let's say you have a geometry problem where you need to find the exact side length of a square with area 72, or simply simplify the radical √72.

  1. 1 Enter 72 in the input field.
  2. 2 Click Calculate.
  3. The Result: Decimal: 8.48528137 Simplified Radical: 6√2 (since 72 = 36 × 2 and √36 = 6).

How This Calculator Works

This tool combines standard arithmetic with a simplification algorithm to give you the most useful answer possible. Here is the logic it follows:

1. Calculation

It computes the principal square root of your input. If the result is a whole integer (like 6 for √36), it identifies it as a Perfect Square.

2. Simplification

If the result is not a whole number, the calculator checks for hidden perfect square factors. For instance, √50 isn't a whole number, but it contains 25. The tool factors this out to present the simplified form 5√2.

3. Decimal Approximation

Finally, for non-perfect squares, it calculates the precise decimal value up to 8 decimal places (e.g., 7.07106781) for practical use in measurements or engineering.

Key Square Root Formulas

Definition

y = √x  ⇔  y² = x

The inverse operation of squaring a number.

Product Rule (Simplification)

√(a × b) = √a × √b

Essential for breaking down radicals (e.g., √50 → √25 × √2).

Quotient Rule

√(a / b) = √a / √b

Allows splitting roots across fractions.

Negative Numbers

√(-x) = i√x

Where i is the imaginary unit, defined as i² = -1.

First 10 Perfect Squares

Memorizing these common roots helps with mental math and estimation.

Number (x) Square Root (√x) Equation
1 1 1 × 1 = 1
4 2 2 × 2 = 4
9 3 3 × 3 = 9
16 4 4 × 4 = 16
25 5 5 × 5 = 25
36 6 6 × 6 = 36
49 7 7 × 7 = 49
64 8 8 × 8 = 64
81 9 9 × 9 = 81
100 10 10 × 10 = 100

Understanding Square Roots

The concept of a square root is fundamental to algebra and geometry. Historically, it originated from a simple geometric problem: if you know the area of a square, what is the length of its side? For example, a square plot of land with an area of 100 square meters has sides exactly 10 meters long, because \(10 \times 10 = 100\).

Rational vs. Irrational Roots

Numbers like 9, 16, and 25 are called perfect squares because their square roots are whole integers (3, 4, and 5). However, most numbers do not factor so cleanly. The square root of 2, for instance, is an irrational number. Its decimal form (approx. 1.414213...) continues forever without repeating. In mathematics and engineering, it is often more precise to leave these values in "radical form" (like \(\sqrt{2}\)) rather than rounding them to a decimal.

Why "Principal" Roots Matter

Mathematically, the equation \(x^2 = 25\) has two solutions: \(5\) and \(-5\). However, the radical symbol (\(\sqrt{}\)) specifically denotes the principal (non-negative) square root. This is why \(\sqrt{25}\) is simply 5. If you need to express both solutions, you must use the plus-minus sign (\(\pm\sqrt{25}\)).

Learn More from Trusted Sources

Frequently Asked Questions

What does this Square Root Calculator do?
This accurate, online square root calculator is a free math tool that helps you quickly compute the square root of any value. It can determine decimal results, simplify radical expressions, show complex roots for negative inputs, and even generate batch tables. The user-friendly interface is designed so any user can access advanced functions without needing to know the underlying algorithm or formula.

How do I use the basic Calculator mode?
In Calculator mode, enter a number in the input field (for example, 50 or −4), choose the desired precision for your result, and click Calculate. The app will compute the output in real time, display the main square root, and, when possible, show a simplified radical form. You can also toggle options to display both positive and negative roots or view step-by-step math operations used by the calculator.

Can the calculator simplify radicals like √72?
Yes. This mathematical tool can simplify radical expressions as part of the same operation that finds the square root. For example, when you enter 72 and calculate, the interface shows the decimal output (≈ 8.48528137) and the simplified radical 6√2. Internally, the algorithm processes the number into factors (72 = 36 × 2) and uses a precise formula to express the result in simplest form.

What is the “Show Negative Root” option for?
By default, the calculator displays the principal (positive) square root. When you enable Show Negative Root, it will display both +√x and −√x, written as ±. This is important when you solve an equation like x² = 16, where the correct mathematical result includes two values (x = 4 and x = −4). The function helps you see every valid solution in a single, clear output.

How precise are the decimal results?
You can set how many digits the app should display using the precision control. The calculator is designed to compute and round the result to the specified number of decimal places, giving you a precise and reliable numerical value for your square root. This makes it efficient for homework, exams, and any calculation where accuracy matters.

Can the calculator handle negative numbers and complex roots?
Yes. When you input a negative number, the calculator uses an advanced algorithm to compute the corresponding complex square root. The result is displayed using the imaginary unit, so an expression like √(−4) is processed correctly and shown in standard mathematical form. This is especially useful if you are working with equations or formulas in higher-level math.

What is the Table Generator or Batch mode used for?
The Table Generator mode lets you load and process many values at once. You can generate a table of numbers and their square roots—for example, from 1 to 100—in a fast, responsive way. This feature is ideal for teachers preparing reference sheets, students revising math formulas, or any user who needs a quick lookup table from a single online tool.

What is the Radical Toolbox and when should I use it?
The Radical Toolbox is an advanced function for working with more complex expressions, not just a single square root. You can use it to simplify expressions or rationalize denominators, such as turning 1/√3 into a cleaner mathematical form. Simply enter the expression, and the calculator will process the operation and display a step-by-step solution that follows standard algebraic rules.

Can I see step-by-step explanations of the simplification?
Yes. When you enable Show Steps, the app displays how each expression is solved. For example, with √72 it shows how the number is broken into factors, how the formula for square roots is applied, and how the final simplified expression 6√2 is obtained. This makes the calculator a helpful learning tool for understanding the math behind each equation, not just the final output.

Who is this Square Root Calculator best suited for?
This online, mobile-friendly square root calculator is designed for anyone who needs a fast, intuitive, and user-friendly way to compute roots: students, teachers, tutors, and professionals. Whether you’re solving equations, checking homework, or exploring advanced expressions, the responsive interface and real-time results make it a reliable app for everyday mathematical work.

Other Precise Calculators

Factoring Calculator

Find factors, GCF, and LCM instantly.

Rounding Calculator

Round numbers to decimals or sig figs.

Fraction Calculator

Add, subtract, and multiply fractions.

BMR Calculator

Calculate Basal Metabolic Rate.

Discount Calculator

Calculate percentage savings and price.

Tip Calculator

Calculate gratuity and split the bill.

ROI Calculator

Calculate investment return percentage.

Age Calculator

Calculate exact age in years and days.

Calorie Calculator

Estimate daily calories for weight goals.