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Indices, SMB-032

Rationalise the denominator of the surd fraction  `(sqrt(12))/(sqrt(6)-2)`.   (3 marks)

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`3sqrt(2)+2sqrt(3)`

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`(sqrt(12))/(sqrt(6)-2)` `=(2sqrt(3))/(sqrt(6)-2) xx (sqrt(6)+2)/(sqrt(6)+2)`  
  `=(2sqrt(3)(sqrt(6)+2))/((sqrt(6))^2-2^2)`  
  `=(2sqrt(18)+4sqrt(3))/(2)`  
  `=(6sqrt(2)+4sqrt(3))/(2)`  
  `=3sqrt(2)+2sqrt(3)`  

Indices, SMB-031

Rationalise the denominator of the surd fraction  `(8-2sqrt(6))/(3sqrt(2)+2sqrt(3))`.   (3 marks)

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`6sqrt(2)-14/3sqrt(3)`

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`(8-2sqrt(6))/(3sqrt(2)+2sqrt(3))`

`=(8-2sqrt(6))/(3sqrt(2)+2sqrt(3))xx(3sqrt(2)-2sqrt(3))/(3sqrt(2)-2sqrt(3))`

`=((8-2sqrt(6))(3sqrt(2)-2sqrt(3)))/((3sqrt(2))^2-(2sqrt(3))^2)`

`=(24sqrt(2)-16sqrt(3)-6sqrt(12)+4sqrt(18))/(18-12)`

`=(24sqrt(2)-16sqrt(3)-12sqrt(3)+12sqrt(2))/6`

`=(36sqrt(2)-28sqrt(3))/6`

`=6sqrt(2)-14/3sqrt(3)`

Indices, SMB-026

Find `a`  and  `b`  such that  `a,b`  are real numbers and

`(6sqrt3-sqrt5)/(2sqrt5)= a + b sqrt15`.  (2 marks)

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`a= -1/2, \ b=3/5`

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`(6sqrt3-sqrt5)/(2sqrt5)` `=(6sqrt3-sqrt5)/(2sqrt5) xx (sqrt5)/(sqrt5)`  
  `=(sqrt5(6sqrt3-sqrt5))/(2 xx5)`  
  `=(6sqrt15-5)/10`  
  `=-1/2 + 3/5 sqrt15`  

  
`:. a= -1/2, \ b=3/5`

Indices, SMB-025

Show working to find `a` and `b` such that  `a,b`  are real numbers and

`(sqrt32-6)/(3sqrt2) = a + bsqrt2`.  (2 marks)

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`:. a = 4/3, \ b = -1`

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`(sqrt32-6)/(3sqrt2) xx (sqrt2)/(sqrt2)` `= (sqrt2(4sqrt2-6))/6`
  `= (8-6sqrt2)/6`
  `= 4/3-sqrt2`

 
`:. a = 4/3, \ b = -1`

Indices, SMB-024

Show working to simplify `a` and `b` such that  `a, b`  are real numbers and

`(8-sqrt27)/(2sqrt3) = a + bsqrt3`.   (2 marks)

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`:. a =-3/2, \ b = 4/3`

Show Worked Solution
`(8-sqrt27)/(2sqrt3) xx (sqrt3)/(sqrt3)` `=(sqrt3(8-3sqrt3))/(2xx3)`
  `= (8sqrt3-9)/6`
  `= -3/2 + 4/3sqrt3`

 
`:. a = -3/2, \ b = 4/3`