Surds
2026-02-28 09:05 Diff
https://brightchamps.com/en-us/math/numbers/surds

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Last updated on December 2, 2025

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 their various types.

What are Surds?

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A 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 to 2. When we simplify √4, we get two because the square root of 4 is 2. Surds help in keeping calculations exact rather than using decimal approximations.
 

Here are some surds examples:
 

\(√2, √3, √5, and √7\)are surds because they cannot be simplified further.
 

√16 is not a surd because it simplifies to 4.
 

Using surds in calculations yields precise results rather than approximate decimals, which is helpful in algebra and higher math.
 

Properties of Surds

Surds are square roots that cannot be simplified into regular numbers. Knowing their rules makes math easier and more fun!
 

  • Adding or subtracting surds with numbers: You can’t turn a number plus a surd into a single surd.

    Example:

    \(3 + √2 stays as 3 + √2.\)

  • When Surds Are Equal: Two surds are equal only if the numbers inside and outside the square root are the same.

    Example:

    If \(p + √q = x + √y,\) then p = x and q = y.

  •  Breaking Down Trickier Surds: Sometimes a surd is inside another square root.

             Example:

             \(√(9 + √16) = 3 + 2, so √(9 − √16) = 3 − 2.\)

  • Adding and subtracting like surds: You can only add or subtract surds that are the same.

            Example:

          \(​​​​​​​2√3 + 5√3 = 7√3,\) but √2 + √3 cannot be combined.

  •  Multiplying Surds: You can learn how to multiply surds easily.
         

  Example:

                       √2 × √8 = √(2 × 8) = √16 = 4.

  •  Dividing Surds: Dividing surds works the same way as multiplying.

            Example:

          \(√8 ÷ √2 = √(8 ÷ 2) = √4 = 2.\)

What are the Types of Surds?

We can classify surds into six different types:

  1. Simple surds: A simple surd has only one term. For example, √7.
     
  2. Pure surds: When surds are completely irrational, we call them pure surds. For example, √3.
     
  3. Similar surds: They are surds with the same radicand. For example, \(√3, 2√3, 5√3\).
     
  4. Mixed surds: Mixed surd is a product of rational and irrational numbers. For example, 6√3 is a mixed surd because 6 is a rational number, while √3 is irrational.
     
  5. 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.
     
  6. Binomial surds: 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.

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What are the Rules for Surds?

There are 6 rules that we use for the calculation of surds: 

Rule 1: \(\sqrt{a\times b} = \sqrt a \times \sqrt b\)

Example: \(\sqrt{36} = \sqrt{9×4} = \sqrt9 × \sqrt4 = 3 × 2 = 6\)

Rule 2: \({\sqrt a \over \sqrt b} = \sqrt{a \over b}\)

Example: \({\sqrt{18} \over \sqrt2} = {\sqrt{18\over2}} = \sqrt9 = 3\)

Rule 3: \({b\over \sqrt a} = {{b\over \sqrt a} \times {\sqrt a\over \sqrt a}} = {b \sqrt a \over \ a}\)

Example: \({3\over \sqrt 5} = {{3\over \sqrt 5} \times {\sqrt 5\over \sqrt 5}} = {3 \sqrt 5 \over \ 5}\)

Rule 4: \(a \sqrt c \space \pm \space b\sqrt c = (a \space \pm \space b)\times \sqrt c\)

Example: \(5 \sqrt 5 \space \pm \space 3\sqrt 5 = (5 \space \pm \space 3)\times \sqrt 5\)

Rule 5: \({c \over {a\space+\space b\sqrt n}} = {{c \over {a\space+\space b\sqrt n}}} \times {{a\space-\space b\sqrt n}\over {a\space-\space b\sqrt n}} = {c \times ({a\space-\space b\sqrt n)}\over {a^2\space-\space b^2 n}}\)

Example: \({5\over {4\space+\space 2\sqrt 3}} = {{5 \over {4\space+\space 2\sqrt 3}}} \times {{4\space-\space 2\sqrt 3}\over {4\space-\space 2\sqrt 3}} = {5 \times ({4\space-\space 3\sqrt 2)}\over {4^2\space-\space (2^2 \times 3)}} = {20-15 \sqrt 2\over 4}\)

Rule 6: \({c \over {a\space-\space b\sqrt n}} = {{c \over {a\space-\space b\sqrt n}}} \times {{a\space+\space b\sqrt n}\over {a\space+\space b\sqrt n}} = {c \times ({a\space+\space b\sqrt n)}\over {a^2\space-\space b^2 n}}\)

Example: \({5\over {4\space-\space 2\sqrt 3}} = {{5 \over {4\space-\space 2\sqrt 3}}} \times {{4\space+\space 2\sqrt 3}\over {4\space+\space 2\sqrt 3}} = {5 \times ({4\space+\space 3\sqrt 2)}\over {4^2\space-\space (2^2 \times 3)}} ={20+15 \sqrt 2\over 4} \)

How to Solve Surds?

When solving for surds, there are a few steps we need to look out during each operation:
 

  • Addition or subtraction: Only like surds can be combined. Example: 3√5 + 7√5 = 10√5.
  • Multiplication: Multiply the radicands inside the root. Example: \(\sqrt 3 \times \sqrt {12} = \sqrt{36} = 6\)
  • Division: Divide the radicands before simplifying the root. Example: \(\sqrt{18} \space/ \sqrt2 = \sqrt 9 = 3\)

These are some of the few ways we can solve for surds when using the basic arithmetic operations. 

Surds WorkSheet:

Identify Surds 

√5

√16

√2

√49

√11

Trips and Tricks to Master Surds

Working with surds can feel tricky at first, but with a few simple tips, it becomes much easier. These tricks are also helpful for parents and teachers to guide children and simplify surds in a fun, easy way.

  • Memorize perfect squares: Start by recalling the perfect squares, such as 4, 9, 16, 25, and so on. Knowing these helps you calculate quickly and simplifies the simplification of surds.
  •  Break large numbers into factors: When finding the square root of a big number, break it down into smaller factors and pair them. Take one number from each pair and move it outside the square root. Any leftover numbers stay inside.

            Example:\(√180 = √(2 × 2 × 3 × 3 × 5) = (2 × 3)√5 = 6√5\)

           This trick makes simplification of surds easier for kids and for indies learning on their own.

  •  Rationalizing the denominator: Sometimes a surd appears in the denominator of a fraction. To simplify, multiply both the top and the bottom of the fraction by the denominator.

    Example:

    \( √4 / √5 = (√4 × √5) / (√5 × √5) = √20 / 5\)

  •  Remember Important Formulas: Knowing square and cube formulas makes calculations faster and easier. These formulas also help in the simplification of surds:

    \((a + b)² = a² + b² + 2ab\)

    \((a − b)² = a² + b² − 2ab\)

    \(a³ + b³ = (a + b)(a² − ab + b²)\)

    \(a³ − b³ = (a − b)(a² + ab + b²)\)

  •  Adding or subtracting like surds: You can only add or subtract like surds. Just combine the numbers outside the square root.

    Example:

    \(a√n ± b√n = (a ± b)√n\)

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:

Real-Life Applications of 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.
     
  • Computer graphics: Surds are used to model 3D shapes in a two-dimensional space to program precise movements of objects in animations and graphics.
     
  • Physics: Surds are widely applied in the field of physics such as optics and waves, to design lenses and studying sound waves.

Problem 1

Simplify √72.

Okay, lets begin

6√2

Explanation

Factorize 72, and you will get \({\sqrt{36 \times 2}} = \sqrt{36} \times \sqrt 2\)

Since \(\sqrt{36}\) = 6 we get 6√2.

Well explained 👍

Problem 2

Add 3√5 +7√5

Okay, lets begin

10√5

Explanation

Since both terms have the same surd √5, we will add the coefficients: \(3 + 7 = 10\)


So, \(3√5 + 7√5 = 10√5\)

Well explained 👍

Problem 3

8√3 - 2√3

Okay, lets begin

6√3

Explanation

So, \(8√3 - 2√3 = 6√3\)

Well explained 👍

Problem 4

Divide: √48/√3

Okay, lets begin

4

Explanation

\({\sqrt {48} \over \sqrt {3} } = {\sqrt{48\over 3}} = \sqrt {16} = 4\)

Well explained 👍

Problem 5

Solve: (√18 + √8) / √2

Okay, lets begin

5

Explanation

\(\sqrt {18} = 3\sqrt 2, \sqrt8 = 2\sqrt 2\)


Sum of \(\sqrt {18} \space\text{and} \sqrt8 \) : \(x3√2 + 2√2 = 5√2\)


Dividing by\( √2: 5√2 ÷ √2 = 5\)

Well explained 👍

FAQs on Surds

1.Why are surds irrational?

Surds are non-repeating, non-terminating decimal expansions, which is why surds are irrational.

2.How can we simplify a surd?

We can simplify a surd by factorizing the number under the root sign to extract any perfect squares. For example, \(\sqrt {50} = \sqrt{25 \times 2} = 5\sqrt2\) 

3.What does it mean to rationalize a denominator?

We rationalize a denominator by rewriting a fraction so that the denominator does not contain a surd. In 1/√2, we rationalize the denominator by multiplying the numerator and denominator by √2, which gives √2/2. 

4.What are like and unlike surds?

Like surds have the same radicand (number that is inside the square root). Unlike surds have different radicands such as 2√3.

5.Can surds be negative?

No, a surd itself is always a positive number because square roots of positive numbers are positive. However, an expression involving a surd can be negative, but the surd itself remains positive.

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.