# Simplify Radical Calculator

Simplify square roots by factoring out perfect squares. See prime factorization, simplified radical form, and decimal value. Free radical simplification calculator.

## What this calculates

Simplify any square root by factoring out perfect squares. Enter the number under the radical sign and get the simplified form, prime factorization, and decimal approximation.

## Inputs

- **Number Under the Radical (√n)** — min 1 — The number inside the square root sign.

## Outputs

- **Simplified Form** — formatted as text — The simplified radical expression.
- **Prime Factorization** — formatted as text — The prime factorization of the radicand.
- **Decimal Approximation** — The decimal value of the square root.
- **Is Perfect Square?** — formatted as text — Whether the number is a perfect square.
- **Simplification Steps** — formatted as text — Step-by-step breakdown of the simplification.

## Details

Simplifying a radical means rewriting a square root in its simplest form by extracting any perfect square factors.

**The Process:**

1. Find the prime factorization of the number under the radical.
2. Group the prime factors into pairs.
3. For each pair, pull one copy outside the radical.
4. Whatever remains inside stays under the radical sign.

**Worked Example: sqrt(72)**

1. Prime factorization: 72 = 2³ x 3² = 2 x 2 x 2 x 3 x 3
2. Pairs: one pair of 2s, one pair of 3s, one leftover 2
3. Pull out: 2 x 3 = 6
4. Remaining inside: 2
5. Result: sqrt(72) = 6sqrt(2)

Check: 6² x 2 = 36 x 2 = 72.

**Another Example: sqrt(200)**

200 = 2³ x 5² = (2²)(5²)(2). Pull out 2 x 5 = 10, leave 2 inside. sqrt(200) = 10sqrt(2).

**Perfect Squares:**

If all prime factors come in pairs, the number is a perfect square and the radical simplifies to a whole number. sqrt(144) = 12 because 144 = 2⁴ x 3² = (2² x 3)² = 12².

**Why simplify radicals?** Simplified form makes it easier to compare, add, and multiply radical expressions. In algebra, teachers expect answers in simplified radical form rather than as decimals.

## Frequently Asked Questions

**Q: How do I simplify a square root?**

A: Find the largest perfect square that divides the number. Factor it out: sqrt(50) = sqrt(25 x 2) = 5sqrt(2). Or use prime factorization: 50 = 2 x 5². The pair of 5s comes out, leaving 2 inside: sqrt(50) = 5sqrt(2).

**Q: What is a perfect square?**

A: A perfect square is an integer whose square root is also an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on. If the radicand is a perfect square, the radical simplifies completely to a whole number.

**Q: Can all square roots be simplified?**

A: Not all. If the number under the radical is prime (like 2, 3, 5, 7, 11) or has no repeated prime factors, the radical is already in simplest form. sqrt(7) and sqrt(30) cannot be simplified further. Only numbers with perfect square factors can be simplified.

**Q: How do I add or subtract radicals?**

A: You can only add or subtract radicals that have the same radicand. For example, 3sqrt(5) + 2sqrt(5) = 5sqrt(5). But 3sqrt(5) + 2sqrt(3) cannot be combined. Always simplify first: sqrt(12) + sqrt(27) = 2sqrt(3) + 3sqrt(3) = 5sqrt(3).

**Q: What about cube roots and higher?**

A: The same idea applies to cube roots but you look for groups of three instead of pairs. For cube root of 54: 54 = 2 x 3³. The triple of 3s comes out: cbrt(54) = 3 cbrt(2). This calculator handles square roots; for cube roots, group prime factors into triples.

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Source: https://vastcalc.com/calculators/math/simplify-radical
Category: Math
Last updated: 2026-04-08