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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)`

Show Worked Solution
`(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)`

Show Worked Solution

`(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-029

Expand and simplify  `(4sqrt(3)+sqrt(2))(4sqrt(8)-sqrt(12))`.   (2 marks)

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`30sqrt(6)-8`

Show Worked Solution

`(4sqrt(3)+sqrt(2))(4sqrt(8)-sqrt(12))`

`=16sqrt(24)-4sqrt(36)+4sqrt(16)-sqrt(24)`

`=15sqrt(4xx6)-24+16`

`=30sqrt(6)-8`

Indices, SMB-028

Expand and simplify  `(sqrt(20)+2sqrt(10))(3sqrt(6)-sqrt(3))`.   (2 marks)

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`4sqrt(30)+10sqrt(15)`

Show Worked Solution

`(sqrt(20)+2sqrt(10))(3sqrt(6)-sqrt(3))`

`=3sqrt(120)-sqrt(60)+6sqrt(60)-2sqrt(30)`

`=3sqrt(4xx30)+5sqrt(4xx15)-2sqrt(30)`

`=6sqrt(30)+10sqrt(15)-2sqrt(30)`

`=4sqrt(30)+10sqrt(15)`

Indices, SMB-030

Find the reciprocal of   `1/a + 1/b -c/(ab)`.   (3 marks)

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`(ab)/(a+b-c)`

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`1/a + 1/b -c/(ab)` `=b/(ab)+a/(ab)-c/(ab)`
  `=(b+a-c)/(ab)`

 
`text(Reciprocal of)\ \ x = x^(-1)`

`:.\ text(Reciprocal of)\ \ (b+a-c)/(ab)=>((b+a-c)/(ab))^(-1)=(ab)/(a+b-c)`

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`

Show Worked Solution
`(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`

Show Worked Solution
`(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`

Indices, SMB-022

Simplify  `((4p^2)/8)^(-3)`  and express as a fraction involving non-negative indices only.  (3 marks)

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`8/(p^6)`

Show Worked Solution
`((4p^2)/8)^(-3)` `= ((p^2)/2)^(-3)`
  `=p^(2xx -3)/(2^(-3))`
  `= (p^(-6))/(2^(-3))= (2^(3))/(p^(6))`
  `= 8/(p^6)`

Indices, SMB-021

Simplify  `((3y^3)/2)^(-2)`  and express as a fraction involving non-negative indices only.  (3 marks)

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`4/(9y^6)`

Show Worked Solution
`((3y^3)/2)^(-2)` `=(3^(-2)xxy^((3xx-2)))/(2^(-2))`
  `= (2^(2)xxy^(-6))/(3^(2))`
  `= 4/(9y^6)`

Indices, SMB-020

Express  `4a^(-5) -: 12a^4`  as a fraction involving non-negative indices only.  (1 mark)

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`1/(3a^9)`

Show Worked Solution
`4a^(-5) -: 12a^4` `=(4a^(-5))/(12a^4)`
  `= a^((-5-4))/3`
  `= (a^(-9))/3`
  `=1/(3a^9)`

Indices, SMB-004 MC

Which expression is equal to  `6^3 xx 36^2`?

  1. `6 xx 3 xx 36 xx 2`
  2. `6 xx 6 xx 6 xx 6 xx 6`
  3. `6 xx 6 xx 6 xx 36 xx 6`
  4. `6 xx 6 xx 6 xx 6 xx 6 xx 6 xx 6`
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`D`

Show Worked Solution
`6^3 xx 36^2` `= (6xx6xx6) xx (6xx6)^2`
  `= (6xx6xx6) xx (6xx6) xx (6 xx6)`
  `=6 xx 6 xx 6 xx 6 xx 6 xx 6 xx 6`

 
`=>D`

Functions, 2ADV F1 2004 HSC 1d

 Find integers  `a`  and  `b`  by showing working to expand and simplify 

`(3-sqrt2)^2 = a-b sqrt2`.  (2 marks)

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`a = 11,\ b = 6`

Show Worked Solution
`(3-sqrt2)^2` `= 9-6 sqrt2 + (sqrt2)^2`
  `= 9-6 sqrt2 + 2`
  `= 11-6 sqrt2`
   
`:.\ a = 11, \ \ b = 6`

Algebra, 2UG 2010 HSC 27a

Fully simplify  `(4x^2)/(3y) -: (xy)/5`.   (3 marks) 

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 `(20x)/(3y^2)`

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♦♦ Mean mark 33%.
MARKER’S COMMENT: Remember to “invert and multiply” when dividing a fraction by a fraction (see Worked Solution).
`(4x^2)/(3y) -: (xy)/5` `= (4x^2)/(3y) xx 5/(xy)`
  `= (20x^2)/(3xy^2)`
  `= (20x)/(3y^2)`

Algebra, 2UG 2011 HSC 12 MC

Which of the following expresses `(6x^2y)/3-:(2y)/5` in its simplest form?

  1. `5x^2`
  2. `30x^2y`
  3. `1/(5x^2)`
  4. `5/(4x^2y^2)`
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`A`

Show Worked Solution
ALGEBRA TIP: When you divide by a fraction, invert the divisor and multiply as shown in the Worked Solution.
`(6x^2y)/3-:(2y)/5` `=(6x^2y)/3xx5/(2y)`
  `=(30x^2y)/(6y)`
  `=5x^2`

 
`=>A`

L&E, 2ADV E1 2011 HSC 1c

Solve   `2^(2x+1)=32`.    (2 marks) 

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`x=2`

Show Worked Solutions
MARKER’S COMMENT: Many students correctly solved this by taking the logarithms of both sides.
`2^(2x+1)` `=32`
`2^(2x+1)` `=2^5`
`2x+1` `=5`
`:. x` `=2`