1068 LearnersLast updated on June 18th, 2025
Surds
Surd is a term that we use to refer to square roots of non-perfect squares. Surds also include higher roots, such as cube roots, that cannot be simplified into rational numbers. In this topic, we are going to learn more about surds and its various types.
What are Surds?
Surd is a mathematical term used to describe irrational numbers that can be expressed as the root of an integer. When a root cannot be simplified further, we call that a surd. For example, √4 is not a surd because it can be simplified even further. When we simplify √4, we get 2 because the square root of 4 is 2. Surds help in keeping calculations exact rather than using decimal approximations.
Properties of Surds
When calculating surds, there are a few key properties that you must follow:
- The sum or difference of a rational number and a quadratic surd cannot result in a quadratic surd.
- If p ± √q = x ± √y, then p = x and q = y.
- If √(a+√b)=√c+√d, then √(a-b) =√c-√d, the same goes for the inverse.
- Only like surds can be added or subtracted.
- Surds can be multiplied if standard root rules are followed.
- Surds can be divided. For example, √a/√b=√a/b.
What are the Types of Surds?
We can classify surds into six different types:
- Simple surds: A simple surd has only one term. For example, √3.
- Pure surds: When surds are completely irrational, we call them pure surds. For example, √3.
- Similar surds: They are surds with the same radicand. For example, √3, 2√3, 5√3.
- Mixed surds: Mixed surd is a product of rational and irrational numbers. For example, 63 is a mixed surd because 6 is a rational number, while √3 is irrational.
- Compound Surds: Compound surds are the addition or subtraction of two or more surds. For example √5+√3 is the sum of two different surds.
- Binomial surd: It takes two separate surds to form one binomial surd. For example, √3 + √7 is a binomial surd because it has two surds added together.
What are the Rules for Surds?
There are 6 rules that we use for the calculation of surds:
Rule 1: √a × b = √a × √b
Example: √36 = √3 × 12 = √3 × √12
Rule 2: √a/√b = √ab
Example: 182=182=9 =3
Rule 3: b/√a=(b/√a) × (√a/√a=b√a/√a
Example: 3√5=(3/√5) × (5/√5)=35/√5
Rule 4: a√c ± b√c=(a ± b)c
Example: 5√5 ± 3√5 = (5 ± 3)√5
Rule 5: c/(a + b √n)
Example: 5/(4 + 2√3)
Rule 6: c/(a - b√n)
Example: 5/(4 - 2√3)
How to Solve Surds?
When solving for surds there are a few steps we need to look out during each operation:
- Simplify: We factor out the perfect squares. Example: √72 = 6√2.
- Addition or subtraction: Only like surds can be combined. Example: 3√5+7√5=10√5.
- Multiplication: Multiply the radicands inside the root. Example: √3 × √12 = √36 = 6
- Division: Divide the radicands before simplifying the root. Example: √18/√2 = √9 = 3
These are some of the few ways we can solve for surds when using the basic arithmetic operations.
Real-life Applications on Surds
Surds are used in fields where precise calculations involving irrational numbers are required:
- Engineering and construction: We use surds to calculate problems involving diagonal distances, slopes, and structural designs.
- Finance and banking: Compound interest often includes surds when working with non-repeating decimal growth rates.
- Navigation and GPS systems: Surds are used in distance formulas when calculating precise locations on Earth’s curved surface.
Common Mistakes and How to Avoid Them in Surds
Students tend to make mistakes while learning surds. Being aware of such mistakes can work in our favor. Take a look at some of the most common mistakes and ways to avoid them:
Mistake 1
Incorrectly simplifying surds
Students may write √50 as √25+√2 instead of 5√2. Students must remember that √a×b=√a ×√b. However, √a + b ≠ √a +√b. Always make sure to factorize and take the perfect squares.
Mistake 2
Adding and subtracting unlike surds
Students may think that adding unlike surds is the same as adding like surds. But surds can only be added or subtracted if they have the same radicand (number inside the square root). If they are different, then the surds cannot be simplified further.
Mistake 3
Forgetting to rationalize the denominator
When a fraction has a surd in the denominator, we need to simplify it. This is done by multiplying the numerator and denominator with the surd. This will remove the surd from the denominator.
Mistake 4
Incorrectly multiplying surds
When multiplying, students must make sure they use the correct rule √a×b = √a ×√b. If the correct rule is not applied then it will lead to an incorrect answer.
Mistake 5
Dividing surds incorrectly
When dividing, students must make sure to apply the property √a/√b=√a/b before simplifying. This rule will avoid any incorrect answers
Solved Examples on Surds
Problem 1
Simplify √72.
6√2
Explanation
Factorize 72, and you will get √(36 × 2) = √36 × √2.
Since √36 = 6 we get 6√2.
Problem 2
Add 3√5 +7√5
10√5
Explanation
Since both terms have the same surd √5, we will add the coefficients: 3 + 7 = 10
So, 3√5+7√5=10√5
Problem 3
8√3 - 2√3
6√3
Explanation
So 8√3 - 2√3 = 6√3
Problem 4
Divide: √48/√3
4√3
Explanation
√48/√3 =√48/3 =√16 = 4
Problem 5
Solve: √18 + √8/√2
5
Explanation
√18=3√2, √8=2√2
3√2+2√2=5√2
5√2/√2=5
FAQs on Surds
1.Why are surds irrational?
2.How can we simplify a surd?
3.What does it mean to rationalize a denominator?
4.What are like and unlike surds?
5.Can surds be negative?
6.How can children in United States use numbers in everyday life to understand Surds?
7.What are some fun ways kids in United States can practice Surds with numbers?
8.What role do numbers and Surds play in helping children in United States develop problem-solving skills?
9.How can families in United States create number-rich environments to improve Surds skills?
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
