Introduction to Surds

Surds
In mathematics, a surd is an irrational number that is expressed as a root (like a square root, cube root, etc.) of a rational number, where the root cannot be simplified to a whole number or a rational fraction.

The following diagram shows some laws or rules for surds. Scroll down the page for more examples and solutions for surds.


 

Algebra Worksheets
Practice your skills with the following Algebra worksheets:
Printable & Online Algebra Worksheets

Key Rules and Operations with Surds

1. Multiplication of Surds:
The product of two square roots is the square root of their product.
\(\sqrt{a}\times \sqrt{b}=\sqrt{ab}\)
Example: \(\sqrt{3}\times \sqrt{5}=\sqrt{15}\)
Example: \(2\sqrt{3}\times 4\sqrt{2}=(2\times 4)\sqrt{3\times2} = 8\sqrt{6}\)

2. Division of Surds:
The quotient of two square roots is the square root of their quotient.
\(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\)
Example: \(\frac{\sqrt{18}}{\sqrt{2}}=\sqrt{\frac{18}{2}} =\sqrt{9} = 3\)

3. Addition and Subtraction of Surds:
You can only add or subtract “like” surds (surds with the same number under the root sign). This is similar to combining like terms in algebra.
\(a\sqrt{x}+b\sqrt{x}=(a+b)\sqrt{x}\)
\(a\sqrt{x}-b\sqrt{x}=(a-b)\sqrt{x}\)
Example: \(5\sqrt{7}+3\sqrt{7}=8\sqrt{7}\)
Example: \(10\sqrt{2}-4\sqrt{2}=6\sqrt{2}\)
You cannot add \(\sqrt{2}+\sqrt{3}\)

4. Rationalizing the Denominator
It’s common practice in mathematics to express fractions without surds in the denominator. This process is called rationalizing the denominator.
Check out this lesson on Rationalizing the Denominator

Videos

Surd laws
Two laws of surds or surd rules and how to apply them in some examples to simplify numbers in surd form.

• Show Step-by-step Solutions

Simplifying surds

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Addition and Subtraction of Surds

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Surds - the basics
What are surds and what do you do with them?

• Show Step-by-step Solutions

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