Simplifying Radicals Calculator

Simplify any square root to its simplest form

Enter the number you want to find the simplified square root of (the radicand).

√
Other Titles
Understanding How to Simplify Radicals: A Comprehensive Guide
Learn the process of simplifying radicals by factoring out perfect squares, a fundamental skill in algebra.

What Does it Mean to Simplify a Radical?

Simplifying a radical (or simplifying a square root) means rewriting it in a way that there are no more square roots of perfect squares left inside the radical sign (√). The goal is to take out as much as possible from under the square root symbol.
For example, √20 is not in its simplest form because 20 contains a perfect square factor, 4. The simplified form is 2√5.
The Basic Rule
The process relies on the product property of square roots, which states:
√(a b) = √a √b
This allows us to split a radical into the product of two other radicals. We use this by factoring the number under the radical (the radicand) into a perfect square and another number.

Core Concept Example

  • Let's simplify √48.
  • 1. **Find a perfect square that divides 48:** Perfect squares are 1, 4, 9, 16, 25, 36, ... We can see that 4 divides 48.
  • - `√48 = √(4 * 12) = √4 * √12 = 2√12`
  • 2. **Check the new radical (√12):** Can it be simplified further? Yes, 4 is also a factor of 12.
  • - `2√12 = 2 * √(4 * 3) = 2 * (√4 * √3) = 2 * (2 * √3) = 4√3`.
  • 3. **Final Result:** √48 simplifies to 4√3. Note that we could have found the largest perfect square factor (16) from the start: `√48 = √(16 * 3) = √16 * √3 = 4√3`.

Step-by-Step Guide to Using the Calculator

Our calculator finds the largest perfect square factor for you.
How to Use It:

Method: Finding the Largest Perfect Square

  • To simplify a radical manually, the most efficient method is to find the largest perfect square that divides the radicand.
  • Let's simplify √72:
  • 1. **List perfect squares:** 4, 9, 16, 25, 36, 49, ...
  • 2. **Test them:** Does 36 divide 72? Yes, 72 = 36 * 2.
  • 3. **Rewrite the radical:** `√72 = √(36 * 2)`
  • 4. **Apply the product rule:** `√36 * √2`
  • 5. **Simplify:** `6√2`

Real-World Applications of Simplifying Radicals

Simplifying radicals is a foundational skill in algebra and geometry, necessary for solving many types of problems.
Algebra:
Geometry:

Practical Example in Geometry

  • Find the diagonal of a rectangle with sides 5 cm and 10 cm.
  • Diagonal = `√(5² + 10²) = √(25 + 100) = √125`
  • Simplify √125: The largest perfect square factor of 125 is 25.
  • `√125 = √(25 * 5) = √25 * √5 = 5√5` cm.

Common Misconceptions and Correct Methods

Misconception: √(a + b) = √a + √b
This is a very common mistake. The product property √(a*b) = √a * √b does NOT work for addition or subtraction. You cannot distribute a square root over addition or subtraction.
Example: √16 + √9 = 4 + 3 = 7. However, √(16 + 9) = √25 = 5. Clearly, the two are not equal.
Misconception: Not finding the LARGEST perfect square
If you don't use the largest perfect square, you will have to simplify more than once. For √72, if you use 9 as the factor (√72 = 3√8), you then have to simplify √8 again (√8 = 2√2), leading to 3 * 2√2 = 6√2. This works, but it's more steps than finding the largest factor (36) from the start.

Key Takeaways

  • Always look for the largest perfect square factor of the number under the radical.
  • The product rule works for multiplication, not addition.
  • A radical is fully simplified when the number under the radical has no perfect square factors other than 1.

Mathematical Derivation and Examples

The process of simplifying a radical is essentially prime factorization.
Prime Factorization Method for √180

Summary of Steps

  • 1. Find the largest perfect square factor `p²` of the radicand `n` such that `n = p² * r`.
  • 2. Rewrite the radical as `√(p² * r)`.
  • 3. Separate into `√p² * √r`.
  • 4. Simplify to `p√r`.