Surds Calculator

Understanding Surds

What are Surds?

Surds are irrational numbers that contain square roots which cannot be simplified to remove the square root sign completely. They are often used in algebra and geometry to represent exact values of lengths, areas, or volumes that cannot be expressed as rational numbers.

Formula and Representation

A surd is typically represented in the form:

\[a\sqrt{b}\]

Where:

• \(a\) is the coefficient (a rational number)
• \(\sqrt{b}\) is the square root of \(b\) (where \(b\) is not a perfect square)

Simplification of Surds

To simplify a surd, we follow these steps:

1. Identify the largest perfect square factor of the radicand (\(b\))
2. Take the square root of this factor and multiply it by the coefficient (\(a\))
3. The remaining factor under the square root becomes the new radicand

Example Calculation

Let's simplify the surd \(2\sqrt{12}\):

1. Identify the largest perfect square factor of 12: 4
2. Take the square root of 4 (which is 2) and multiply it by the coefficient 2: 2 * 2 = 4
3. The remaining factor under the square root is 12 ÷ 4 = 3
4. Therefore, \(2\sqrt{12} = 4\sqrt{3}\)

Visual Representation

This diagram shows the simplification of the surd \(2\sqrt{12}\) to \(4\sqrt{3}\).