Types of Surds: Simple, Mixed, Pure and Compound Surds
Types of Surds classify radical expressions according to their terms, coefficients and irrational parts. The main categories include simple, pure, mixed, similar, dissimilar and compound surds. Simplifying the radicand first helps identify the correct type and determine whether two surds can be combined.
What Are the Types of Surds?
A radical expression is a surd only when its value is irrational. Thus, √2 is a surd, but √9 = 3 is not. In the simplest form, √12 = √(4 × 3) = 2√3. The simplified form should be used before classifying a surd or comparing it with another surd.
Types of Surds Formula & Tricks
Important Formulas
Multiply the radicands when the surds have the same root index. For example, √3 × √5 = √15.
Take perfect-square factors outside the radical. For example, √18 = √(9 × 2) = 3√2.
Only surds with the same simplified irrational part can be combined. For example, 3√5 + 2√5 = 5√5.
Squaring a square root removes the radical. For example, (√7)² = 7.
Quick Tricks
First remove perfect-power factors from the radicand. The resulting expression may change its apparent type or show that two surds are similar.
For addition or subtraction, compare the simplified radical parts, not just the original radicands.
Types of Surds Concepts
Simple Surds
A simple surd has no addition or subtraction separating it into multiple surd terms. Its coefficient may be 1 or another rational number. Therefore, both √5 and 4√5 are simple surds, although they belong to different pure or mixed categories.
Pure and Mixed Surds
Examples of pure surds are √3 and ∛7. Examples of mixed surds are 2√3 and 5∛2. The expression √12 is simplified as 2√3, so its simplest form is a mixed surd rather than a pure surd.
Similar and Dissimilar Surds
The rational coefficients can differ in similar surds. For example, 2√7 and 5√7 are similar. In contrast, √2 and √3 are dissimilar. Surds can be added or subtracted directly only when they are similar.
Compound or Binomial Surds
Expressions such as √2 + √3 and 4√5 − √7 are compound surds. A compound surd with exactly two terms is often called a binomial surd. Its terms may be similar or dissimilar; √3 + 2√3 is compound before simplification but equals 3√3.
Classifying a Surd Correctly
First extract perfect powers from each radicand. One remaining radical term indicates a simple surd; multiple terms indicate a compound surd. A coefficient of 1 indicates a pure surd, while a different rational coefficient indicates a mixed surd. Comparing the final irrational parts identifies similar and dissimilar surds.
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Types of Surds: Quick Revision
Use the simplified radical form to identify the type and apply operations correctly.
- A surd has an irrational value expressed in exact radical form.
- A simple surd has one radical term; a compound surd has two or more terms.
- A pure surd has coefficient 1 outside the radical, while a mixed surd has another rational coefficient.
- Similar surds have the same simplified irrational part and can be combined.
- Dissimilar surds have different simplified irrational parts and cannot be directly added or subtracted.
- Always simplify expressions such as √12 and √27 before classifying or combining them.
- √12 = 2√3 and √27 = 3√3, so √12 + √27 = 5√3.
Types of Surds FAQs
Is √12 a pure surd or a mixed surd?
√12 = √(4 × 3) = 2√3. Therefore, in simplest form, it is a mixed surd because it has the rational coefficient 2 outside the radical.
Can √8 and √2 be added directly?
Yes, after simplification. Since √8 = 2√2, √8 + √2 = 3√2; the two surds are similar.
What is the difference between a pure surd and a simple surd?
A pure surd has coefficient 1 outside the radical, such as √5. A simple surd has only one radical term, so 3√5 is simple but mixed.
Can dissimilar surds be combined by addition?
No. Surds such as √2 and √3 have different irrational parts, so √2 + √3 cannot be written as a single surd term by combining coefficients.
What type of surd is √2 + √3?
It is a compound or binomial surd because it contains two radical terms. The terms are also dissimilar because their irrational parts are different.
How do you simplify √50 before classifying it?
√50 = √(25 × 2) = 5√2. Thus, its simplest form is a mixed surd and a simple surd.