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Square Root Surds Calculator

Reviewed by Calculator Editorial Team

This square root surds calculator helps you simplify radical expressions and find exact values for square roots of surds. Surds are numbers left in radical form, such as √2, √(3/2), or √(5 + 2√6).

What is Square Root Surds?

A surd is an irrational number that cannot be expressed as a simple fraction. When you take the square root of a surd, you're looking for the simplest radical form of that number. For example, √8 is a surd, but it can be simplified to 2√2.

Formula: √(a²b) = a√b, where a is the largest perfect square factor of b.

Surds are important in mathematics because they represent exact values rather than decimal approximations. This is particularly useful in geometry, algebra, and calculus where precise measurements are required.

How to Simplify Surds

Simplifying surds involves finding the largest perfect square that divides the radicand (the number under the square root). Here's a step-by-step method:

  1. Factorize the radicand into its prime factors.
  2. Identify pairs of identical prime factors.
  3. Take one factor from each pair out of the square root.
  4. Multiply these factors together to form a coefficient.
  5. Leave any remaining prime factors under the square root.

Note: Not all surds can be simplified. For example, √2 is already in its simplest form.

Let's look at an example to illustrate this process.

Examples

Consider the surd √72. To simplify it:

  1. Factorize 72: 72 = 8 × 9 = 2³ × 3²
  2. Identify perfect square factors: 4 (2²) and 9 (3²)
  3. Take one factor from each pair: √4 × √9 = 2 × 3 = 6
  4. Remaining factors: √(2³ × 3²) = √(2 × 3²) = √(6)
  5. Final simplified form: 6√2

So, √72 simplifies to 6√2.

Example: √(18) = √(9 × 2) = 3√2

FAQ

What is the difference between a surd and a rational number?
A surd is an irrational number that cannot be expressed as a simple fraction, while a rational number can be expressed as a fraction of two integers.
Can all surds be simplified?
No, only surds that have perfect square factors can be simplified. For example, √2 cannot be simplified further.
How do I know if a surd is in its simplest form?
A surd is in its simplest form when there are no perfect square factors left under the square root.
What are some common surds?
Common surds include √2, √3, √5, √7, and √10. These numbers are irrational and cannot be expressed as exact fractions.