Cal11 calculator
Search calculator...

Radicand Square Root Calculator

Reviewed by Calculator Editorial Team

Finding square roots is a fundamental mathematical operation with applications in geometry, algebra, and many practical fields. This calculator helps you compute square roots of any radicand, including negative numbers and complex results.

What is a radicand?

The radicand is the number inside a square root symbol (√). For example, in √9, the radicand is 9. The square root of a number is a value that, when multiplied by itself, gives the original radicand.

Square Root Formula

For a radicand a, the square root is a number x such that:

x × x = a

Or in equation form:

x = √a

The square root of a number is always non-negative, even if the radicand is negative. This is because the product of two negative numbers is positive.

How to calculate square root

Calculating square roots can be done using several methods:

1. Using a calculator

The simplest method is to use a calculator, either physical or digital. Our radicand square root calculator above provides an easy way to compute square roots without manual calculations.

2. Prime factorization method

For perfect squares, you can find the square root by factoring the radicand into prime factors and pairing them:

  1. Factor the radicand into prime factors
  2. Pair the prime factors
  3. Take one factor from each pair to find the square root

Example

Find √36:

  1. 36 = 6 × 6 = 2 × 3 × 2 × 3
  2. Pair the factors: (2 × 2) and (3 × 3)
  3. Take one from each pair: 2 and 3
  4. √36 = 6

3. Long division method

For non-perfect squares, you can use the long division method:

  1. Group the digits in pairs from the decimal point
  2. Find the largest number whose square is less than or equal to the first pair
  3. Subtract and bring down the next pair
  4. Repeat the process

Example

Find √2:

  1. 2.000000
  2. 1 × 1 = 1 (subtract to get 1)
  3. Bring down 00, making 100
  4. 14 × 14 = 196 (too big), so 13 × 13 = 169
  5. Subtract to get 31, bring down 00
  6. This process continues to find √2 ≈ 1.414213

Negative radicands

When the radicand is negative, the square root is not a real number. However, in the complex number system, we can find a solution using the imaginary unit i, where i = √-1.

Square Root of Negative Numbers

For a negative radicand a:

√a = √(-b) = i√b

Where b is the absolute value of a.

Example

Find √-16:

  1. √-16 = √(16 × -1) = √16 × √-1 = 4 × i = 4i

Complex numbers

Complex numbers extend the real number system by adding the imaginary unit i. A complex number has the form a + bi, where a is the real part and b is the imaginary part.

Square Root of Complex Numbers

For a complex number z = a + bi, the square roots are:

√z = ±(√[(a + √(a² + b²))/2] + i × sign(b) × √[(√(a² + b²) - a)/2])

This formula gives two complex square roots for any non-zero complex number.

Practical applications

Square roots have many practical applications in various fields:

1. Geometry

Square roots are used to find the length of the hypotenuse in right-angled triangles (Pythagorean theorem).

2. Physics

Square roots appear in equations involving velocity, acceleration, and other physical quantities.

3. Engineering

Square roots are used in calculations involving electrical circuits, structural analysis, and more.

4. Finance

Square roots are used in risk assessment, option pricing, and other financial calculations.

5. Computer Science

Square roots are used in algorithms, cryptography, and other computational applications.

FAQ

What is the difference between a radicand and a square root?
The radicand is the number inside the square root symbol (√). The square root is the result of the square root operation.
Can I find the square root of a negative number?
In the real number system, no. However, in the complex number system, you can find square roots of negative numbers using the imaginary unit i.
What is the square root of zero?
The square root of zero is zero, because 0 × 0 = 0.
How do I calculate the square root of a fraction?
To find the square root of a fraction, take the square root of the numerator and the denominator separately: √(a/b) = √a / √b.
What is the square root of a square root?
The square root of a square root is the original radicand raised to the power of 1/4: √(√a) = a^(1/4).