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Multiplication and Division of Surds

Like addition and subtraction of surds, multiplication and division of surds have two rules:

  1. sqrt{x}  x  sqrt{y}  =  sqrt{xy}
  2. sqrt{x}  ÷  sqrt{y}  =  sqrt{x/y}

 

Examples of multiplication and division of surds

Example 1: sqrt{7}  x  sqrt{5}  =  sqrt{35}

 

Example 2: 2sqrt{3}  x  3sqrt{2}

=  2 x 3 x sqrt{3~*~2}

=  6sqrt{6}

 

Example 3: 2sqrt{3}  x  3sqrt{3}=  6(sqrt{3})^2

=  6 x 3

=  18

 

Example 4: 5sqrt{8}  x  sqrt{6}

=  5sqrt{48}

=  5sqrt{16~*~3}

=  5 x 4sqrt{3}

=  20sqrt{3}

 

Example 5: {4sqrt{3}~~*~~sqrt{18}}/sqrt{12}

=  {4sqrt{3}~~*~~sqrt{9~*~2}}/sqrt{4~*~3}

=  {4sqrt{3}~~*~~3sqrt{2}}/{2sqrt{3}}

=  6sqrt{2}

 

Example 6: sqrt{32}/sqrt{8}

=  sqrt{32/8}

=  sqrt{4}

=  2

 

Example 7: sqrt{96}  ÷  sqrt{16}

=  sqrt{96/16}

=  sqrt{6}

 

Example 8: Using distributive law a(b + c)  =  ab + ac, simplify sqrt{3}(sqrt{2}~-~3sqrt{3})

=  (sqrt{3}~*~sqrt{2})~-~(3sqrt{3}~*~sqrt{3})

=  sqrt{6}  –  9

 

Example 10: sqrt{x}(2sqrt{x}~+~3sqrt{y})

=  2x  +  3sqrt{xy}

 

 

Example 11: 5sqrt{6}(3sqrt{6}~-~3sqrt{2})

=  (5sqrt{6}~*~3sqrt{6})~-~(5sqrt{6}~*~3sqrt{2})

=  15 x 6  –  15sqrt{12}

=  90  –  15sqrt{4~*~3}

=  90  –  30sqrt{3}

 

In the next section we now look at binomial products with surds