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Square Root in Simplest Radical Form Calculator

Reviewed by Calculator Editorial Team

This calculator helps you find the square root of any number in its simplest radical form. Whether you're studying algebra, preparing for exams, or just need a quick reference, this tool provides accurate results and step-by-step explanations.

What is simplest radical form?

The simplest radical form of a square root is when the radicand (the number under the square root symbol) has no perfect square factors other than 1. This means:

  • The radicand cannot be divided by any perfect square (like 4, 9, 16, etc.) other than 1.
  • The coefficient (the number in front of the square root) is an integer.
  • There are no fractions under the square root.

For example, √32 is not in simplest form because 32 can be divided by 16 (a perfect square). The simplest form is 4√2.

Key Formula

To simplify √a:

  1. Factor a into its prime factors.
  2. Pair the prime factors into perfect squares.
  3. Take one factor from each pair out of the square root.
  4. Multiply the factors outside the square root.

How to simplify square roots

Simplifying square roots involves breaking down the radicand into its prime factors and then pairing them into perfect squares. Here's a step-by-step guide:

Step 1: Factor the radicand

Break down the number under the square root into its prime factors. For example, to simplify √72:

  • 72 ÷ 2 = 36
  • 36 ÷ 2 = 18
  • 18 ÷ 2 = 9
  • 9 ÷ 3 = 3
  • 3 ÷ 3 = 1

So, 72 = 2 × 2 × 2 × 3 × 3.

Step 2: Pair the prime factors

Group the prime factors into pairs of the same number. For √72:

  • 2 × 2 (a pair of 2s)
  • 3 × 3 (a pair of 3s)
  • 2 (remaining)

Step 3: Take one from each pair

Take one factor from each perfect square pair and move it outside the square root. For √72:

  • From 2 × 2, take one 2: √(2 × 2) = 2√(2 × 2)
  • From 3 × 3, take one 3: √(3 × 3) = 3√(3 × 3)

Now you have 2 × 3 × √(2 × 2 × 3 × 3) = 6√72.

Step 4: Simplify the remaining radicand

If there's a remaining radicand that's a perfect square, simplify it. In our example, √72 simplifies to 6√4, which is 6 × 2 = 12.

Tip

Remember that 1 is a perfect square (1 × 1), so if the radicand is a prime number, it's already in simplest form.

Examples

Let's look at a few examples to see how this works in practice.

Example 1: √48

  1. Factor 48: 48 = 16 × 3 = 2 × 2 × 2 × 2 × 3
  2. Pair the factors: (2 × 2) × (2 × 2) × 3
  3. Take one from each pair: 2 × 2 × √(2 × 2 × 3 × 3)
  4. Simplify: 4√(4 × 3) = 4√12 = 4 × 2√3 = 8√3

Example 2: √108

  1. Factor 108: 108 = 36 × 3 = 6 × 6 × 3 = 2 × 3 × 2 × 3 × 3
  2. Pair the factors: (2 × 2) × (3 × 3) × 3
  3. Take one from each pair: 2 × 3 × √(2 × 2 × 3 × 3 × 3)
  4. Simplify: 6√(4 × 9 × 3) = 6√108 = 6 × 6√3 = 36√3

Example 3: √200

  1. Factor 200: 200 = 100 × 2 = 10 × 10 × 2 = 2 × 5 × 2 × 5 × 2
  2. Pair the factors: (2 × 2) × (2 × 2) × (5 × 5)
  3. Take one from each pair: 2 × 2 × 5 × √(2 × 2 × 2 × 2 × 5 × 5)
  4. Simplify: 20√(4 × 4 × 25) = 20√400 = 20 × 20 = 400

FAQ

What is the difference between a radical and a square root?

A radical is the symbol (√) used to denote a root, while a square root specifically refers to the second root of a number. For example, √9 is the square root of 9, which equals 3.

Can I simplify √1?

Yes, √1 is already in simplest form because 1 is a perfect square (1 × 1), and the coefficient is 1.

What if the radicand has a negative number?

The square root of a negative number is not a real number. In simplest radical form, we typically work with positive radicands.

How do I simplify √(a/b)?

To simplify √(a/b), separate it into √a/√b, then simplify each square root separately. For example, √(8/2) = √8/√2 = 2√2/√2 = 2.