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Addition and Subtraction of Surds

Addition and subtraction of surds involve a few simple rules:

  1. we can add or subtract surds only when they are in the simplest form, and
  2. we can add or subtract like surds only.

 Let’s look at both these rules one by one. First lets consider sqrt{9}sqrt{4}

Now sqrt{9} = 3 and sqrt{4} = 2,

so sqrt{9}sqrt{4}

= 3 + 2  

=  5

However sqrt{9}sqrt{4} is not sqrt{9~+~4}sqrt{13} = 3.60555

Hence remember sqrt{x}sqrt{y} <>  sqrt{x~+~y}, and 

sqrt{x}sqrt{y} <>  sqrt{x~-~y}

 

Now the second rule states that we can add or subtract like surds only. So lets consider 

sqrt{12}  +  sqrt{27}

=  sqrt{4~*~3}  +  sqrt{9~*~3}

=  2sqrt{3}  +  3sqrt{3}

=  5sqrt{3}

Here we are adding the two surds only when they are alike, i.e. both the surds have sqrt{3}, so we could add them together – exactly like how it is done in Algebra – adding like terms.

Similarly sqrt{90}  –  sqrt{40}

=  3sqrt{10}  –  2sqrt{10}

=  sqrt{10}

 

Examples of addition and subtraction of surds

  1. 2sqrt{3}  +  5sqrt{3}  –  3sqrt{3}

    =  sqrt{3} (2 + 5 – 3)

    =  4sqrt{3}

  2. sqrt{3}  +  2sqrt{3}  –  6sqrt{3} 

    =  sqrt{3} (1 + 2 – 6)

    =  -3sqrt{3}

  3. 3sqrt{2}  +  5sqrt{3}  –  2sqrt{2}  –  sqrt{3}

    =  (3sqrt{2}  –  2sqrt{2})  +  (5sqrt{3}  –  sqrt{3})

    =  sqrt{2}  +  4sqrt{3}

  4. sqrt{18}  +  2sqrt{8}  –  sqrt{72} 

    =  sqrt{9~*~2}  +  2sqrt{4~*~2}  –  sqrt{36~*~2} 

    =  3sqrt{2}  +  4sqrt{2}  –  6sqrt{2} 

    =  sqrt{2}

  5. 3sqrt{45}  +  sqrt{20}  +  sqrt{32} 

    =  3sqrt{9~*~5}  +  sqrt{4~*~5}  +  sqrt{16~*~2} 

    =  9sqrt{5}  +  2sqrt{5}  +  4sqrt{2} 

    =  11sqrt{5}  +  4sqrt{2}

 

Remember: When we simplify a surd to its simplest form, our purpose is to have atleast one perfect square to take out of the square root.