Surds

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.
 

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:

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

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.

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

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: \(\sqrt{72} = 6\sqrt 2\).

• Addition or subtraction: Only like surds can be combined. Example: 3√5 + 7√5 = 10√5.

• Multiplication: Multiply the radicands inside the root. Example: \(\sqrt 3 \times \sqrt {12} = \sqrt{36} = 6\)

• 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

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:
  
  \(a√n ± b√n = (a ± b)√n\)

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.