Square Root Simplest Radical Form Calculator with Radicand

This calculator helps you find the simplest radical form of a square root with radicand. Radical form is a way of expressing square roots that cannot be simplified further. The radicand is the number inside the square root symbol (√).

What is Radical Form?

Radical form refers to the expression of a square root in its simplest form, where the radicand (the number inside the square root) has no perfect square factors other than 1. For example, √18 can be simplified to 3√2, which is in radical form.

The process of simplifying a square root involves factoring the radicand into a product of perfect squares and other factors. The perfect squares are then taken out of the square root, leaving the remaining factors inside.

Formula

To simplify √a to its simplest radical form:

Factor the radicand (a) into its prime factors.
Identify all perfect square factors of a.
Take the square root of the largest perfect square factor and multiply it by the square root of the remaining factors.

How to Simplify Radicals

Simplifying radicals involves several steps to ensure the expression is in its simplest form. Here's a step-by-step guide:

Identify the radicand: The number inside the square root symbol.
Factor the radicand: Break down the radicand into its prime factors.
Group perfect squares: Identify pairs of prime factors that form perfect squares.
Take the square root of perfect squares: For each perfect square factor, take its square root and move it outside the square root symbol.
Combine the results: Multiply the square roots taken out of the symbol with the remaining factors inside.

Note: The radicand must be a positive integer for simplification. If the radicand is negative, the expression can be written as the square root of the absolute value multiplied by the imaginary unit (i).

Examples

Let's look at a few examples to understand how to simplify square roots to their simplest radical form.

Example 1: √32

Factor 32: 32 = 16 × 2 = 4 × 4 × 2
Identify perfect squares: 16 is a perfect square (4²).
Take the square root of 16: √16 = 4.
Combine: 4 × √2 = 4√2.

So, √32 simplifies to 4√2.

Example 2: √50

Factor 50: 50 = 25 × 2 = 5 × 5 × 2
Identify perfect squares: 25 is a perfect square (5²).
Take the square root of 25: √25 = 5.
Combine: 5 × √2 = 5√2.

So, √50 simplifies to 5√2.

FAQ

What is the radicand in a square root?: The radicand is the number inside the square root symbol (√). For example, in √18, the radicand is 18.
How do I simplify a square root to its simplest radical form?: To simplify a square root, factor the radicand into perfect squares and other factors, then take the square root of the perfect squares and multiply them with the remaining factors inside the square root.
Can I simplify the square root of a negative number?: Yes, but it involves imaginary numbers. The square root of a negative number can be expressed as the square root of its absolute value multiplied by the imaginary unit (i).
What if the radicand is a decimal or fraction?: If the radicand is a decimal or fraction, you can simplify it by converting it to a fraction and simplifying the numerator and denominator separately.
Is the simplest radical form always unique?: Yes, the simplest radical form is unique for any given radicand, as it involves factoring into the largest possible perfect squares.