Indices

Indices, SMB-032

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

--- 5 WORK AREA LINES (style=lined) ---

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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-075

Simplify `(p/q)^3 ÷ (pq^(-2))`. (2 marks)

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`(p^2)/q`

Show Worked Solution

`(p/q)^3 ÷ (pq^(-2))`	`= (p^3)/(q^3) ÷ p/(q^2)`
	`= (p^3)/(q^3) xx (q^2)/p`
	`= (p^2)/q`

Indices, SMB-030

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

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

Show Worked Solution

`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-027

Simplify `(a(b^2)^3)/(a^2b)`. (2 marks)

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`b^5/a`

Show Worked Solution

`(a(b^2)^3)/(a^2b)`	`= (ab^6)/(a^2b)`
	`= b^5/a`

Indices, SMB-026

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

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

--- 4 WORK AREA LINES (style=lined) ---

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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)

Show Answers Only

`:. 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-023

Rationalise the denominator of `1/(4sqrt 3)`. (2 marks)

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`sqrt 3/12`

Show Worked Solution

`1/(4sqrt 3) xx (sqrt 3)/(sqrt 3)`	`= (sqrt 3)/(4xx3)`
	`= sqrt 3/12`

Indices, SMB-022

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

--- 5 WORK AREA LINES (style=lined) ---

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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)

--- 4 WORK AREA LINES (style=lined) ---

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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)

--- 2 WORK AREA LINES (style=lined) ---

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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-019

Express `(24p^(-3)q^4)/(8pq^(2))` as a fraction involving non-negative indices only. (1 mark)

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`(3q^2)/p^4`

Show Worked Solution

`(24p^(-3)q^4)/(8pq^(2))`	`=3p^((-3-1))q^((4-2))`
	`= 3p^(-4)q^2`
	`= (3q^2)/p^4`

Indices, SMB-018

Simplify the expression `(36t^4)/(9t^(-2))`. (1 mark)

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`4t^6`

Show Worked Solution

`(36t^4)/(9t^(-2))`	`= 4t^(4-(-2))`
	`= 4t^6`

Indices, SMB-017

Simplify the expression `(16x^5)/(2x^(-3))`. (1 mark)

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`8x^8`

Show Worked Solution

`(16x^5)/(2x^(-3))`	`= 8x^(5-(-3))`
	`= 8x^8`

Indices, SMB-016

Simplify the expression `(12x)^0/4 xx (5x)/3`. (1 mark)

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`(5x)/12`

Show Worked Solution

`(12x)^0/4 xx (5x)/3`	`= 1/4 xx (5x)/3`
	`= (5x)/12`

Indices, SMB-015

Simplify the expression `(20m^0)/n xx n^2/(5m)`. (1 mark)

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`(4n)/m`

Show Worked Solution

`(20m^0)/n xx n^2/(5m)`	`= 20/n xx n^2/(5m)`
	`= (4n)/m`

Indices, SMB-014

Express `4^(-3)` as a fraction. (1 mark)

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`1/64`

Show Worked Solution

`4^(-3)`	`= 1/4^3`
	`= 1/64`

Indices, SMB-013

Simplify the expression `36t^7 -: 27t^3`. (2 marks)

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`(4t^4)/3`

Show Worked Solution

`36t^7 -: 27t^3`	`= (9 xx 4 xxt^7)/(9xx3xxt^3)`
	`= (4t^4)/3`

Indices, SMB-012

Simplify the expression `24a^8 -: 3a^2`. (2 marks)

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`8a^6`

Show Worked Solution

`24a^8 -: 3a^2`	`= (8xx3xxa^8)/(3 xx a^2)`
	`= 8a^6`

Indices, SMB-011

Simplify the expression `(5p^3)^2`. (2 marks)

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`25p^6`

Show Worked Solution

`(5p^3)^2`	`= 5^2 xx (p^3)^2`
	`= 25p^(3xx2)`
	`=25p^6`

Indices, SMB-010

Simplify the expression `(2x^3)^4`. (2 marks)

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`16x^12`

Show Worked Solution

`(2x^3)^4`	`= 2^4 xx (x^3)^4`
	`= 16x^(3xx4)`
	`=16x^12`

Indices, SMB-009

Find `x` given `100^(x-2) = 1000^x`. (2 marks)

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`-4`

Show Worked Solution

`100^(x-2)`	`= 1000^x`
`(10^2)^(x-2)`	`= (10^3)^x`
`10^(2x-4)`	`= (10)^(3x)`
`2x-4`	`=3x`
`:. x`	`= -4`

Indices, SMB-008

Solve the equation `2^(3x-3) = 8^(2-x)` for `x`. (2 marks)

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`3/2`

Show Worked Solution

`2^(3x-3)`	`= 2^(3(2-x))`
`3x-3`	`= 6-3x`
`6x`	`= 9`
`:. x`	`= 3/2`

Indices, SMB-007

Solve the equation `3^(-4x) = 9^(6-x)` for `x.` (2 marks)

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`-6`

Show Worked Solution

`3^(-4x)`	`= (3^2)^(6-x)`
`3^(-4x)`	`=3^(12-2x)`
` -4x`	`= 12-2x`
`2x`	`=-12`
`:. x`	`=-6`

Indices, SMB-006 MC

Select the expression that is equivalent to `(4^(-3) ÷ 4^6)^3`

`4^(-27)`
`4^(-9)`
`4^(-54)`
`4^(-6)`

Show Answers Only

`A`

Show Worked Solution

`(4^(-3) -: 4^6)^3`	`= (4^(-3-6))^3`
	`= (4^(-9))^3`
	`= 4^(-9 xx 3)`
	`= 4^(-27)`

`=>A`

Indices, SMB-005 MC

Which of the following has the same value as `7^(-11)?`

`-7^(-11)`
`7 xx (-11)`
`1/-7^11`
`1/7^11`

Show Answers Only

`D`

Show Worked Solution

`7^(-11)=1/(7^11)`

`=>D`

Indices, SMB-004 MC

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

`6 xx 3 xx 36 xx 2`
`6 xx 6 xx 6 xx 6 xx 6`
`6 xx 6 xx 6 xx 36 xx 6`
`6 xx 6 xx 6 xx 6 xx 6 xx 6 xx 6`

Show Answers Only

`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`

Indices, SMB-003 MC

`7 xx 2^3 =`

`28`
`42`
`56`
`2744`

Show Answers Only

`C`

Show Worked Solution

`7 xx 2^3`	`= 7 xx 2 xx 2 xx 2`
	`= 56`

`=>C`

Inices, SMB-002 MC

`30^2` is equal to which of the following?

`60^2 ÷ 3`
`3^2 xx 2 xx 5 xx 2 xx 5`
`9 xx 5^2`
`3 xx 10 xx 10`

Show Answers Only

`B`

Show Worked Solution

`30^2 = 900`

`3^2 xx 2 xx 5 xx 2 xx 5`

`= 3^2 xx 10 xx 10`

`= 900`

`=>B`

Indices, SMB-001 MC

Which of the following is equal to 32?

`2^3 xx 2^2`
`2^3 + 2^2`
`3^2 + 2^2`
`3^2 xx 2^2`

Show Answers Only

`=>A`

Show Worked Solution

`2^3 xx 2^2`	`= 8 xx 4`
	`= 32`

`=>A`

L&E, 2ADV E1 2019 HSC 3 MC

What is the value of `p` so that `(a^2a^(-3))/sqrt a = a^p`?

`-3`
`-3/2`
`-1/2`
`12`

Show Answers Only

`B`

Show Worked Solution

`(a^2 a^(-3))/a^(1/2)`	`= a^(-1) xx a^(-1/2)`
	`= a^(-3/2)`

`=> B`

Algebra, 2UG 2016 HSC 26b

Simplify `(8x^4y)/(24x^3y^5)`. (2 marks)

Show Answers Only

`x/(3y^4)`

Show Worked Solution

`(8x^4y)/(24x^3y^5)`	`=(x^((4-3))y^((1-5)))/3`
	`=(xy^(-4))/3`
	`=x/(3y^4)`

Algebra, 2UG 2015 HSC 11 MC

Which of the following is `3x^0 + 5x` in its simplest form?

`6x`
`8x`
`1 + 5x`
`3 + 5x`

Show Answers Only

`D`

Show Worked Solution

♦ Mean mark below 43%. BE CAREFUL!

`3x^0 + 5x`	`= 3 xx 1 + 5x`
	`= 3 + 5x`

`=> D`

Algebra, 2UG 2006 HSC 23a

Simplify `(ab^2)/w xx (4w)/(3b).` (2 marks)

Show Answers Only

`(4ab)/3`

Show Worked Solution

`(ab^2)/w xx (4w)/(3b)`	`=(4ab^2w)/(3bw)`
	`=(4ab)/3`

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)

--- 4 WORK AREA LINES (style=lined) ---

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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 2005 HSC 7 MC

Simplify `2m^2 × 3m p^2`

`5m^2p^2`
`5m^3p^2`
`6m^2p^2`
`6m^3p^2`

Show Answers Only

`D`

Show Worked Solution

`2m^2 × 3m p^2`	`= 6m^((2+1))p^2`
	`=6m^3p^2`

`=> D`

Algebra, 2UG 2008 HSC 1 MC

Which expression is equivalent to `12k^3 ÷ 4k`?

`3k^2 `
`3k^3`
`8k^2`
`8k^3`

Show Answers Only

`A`

Show Worked Solution

`12k^3 -: 4k`	`=(12k^3)/(4k)`
	`=3k^2`

`=> A`

Functions, 2ADV F1 2008 HSC 1e

Expand and simplify `(sqrt3-1)(2 sqrt3 + 5)`. (2 marks)

Show Answers Only

`1 + 3 sqrt 3`

Show Worked Solution

`(sqrt 3-1)(2 sqrt 3 + 5)`

`= 2 xx 3 + 5 sqrt 3-2 sqrt 3-5`

`= 1 + 3 sqrt 3`

Algebra, 2UG 2014 HSC 11 MC

Simplify `6w^4 xx 1/3 w^2`.

`2w^6`
`2w^8`
`18w^6`
`18w^8`

Show Answers Only

`A`

Show Worked Solution

`6w^4 xx 1/3w^2`	`= (6w^4w^2)/3`
	`= 2w^6`

`=> A`

Algebra, 2UG 2010 HSC 27a

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

--- 5 WORK AREA LINES (style=lined) ---

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`(20x)/(3y^2)`

Show Worked Solution

♦♦ 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 2012 HSC 28b

Simplify fully `(18ab)/(3a^2) xx c/b`. (2 marks)

Show Answers Only

`(6c)/a`

Show Worked Solution

♦ Mean mark 35%!

`(18ab)/(3a^2) xx c/b`	`= (18abc)/(3a^2b)`
	`=(6c)/a`

Algebra, 2UG 2011 HSC 12 MC

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

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

Show Answers Only

`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`

Algebra, 2UG 2013 HSC 5 MC

Which of the following is `(12T^2)/(3Txx2W)` in its simplest form?

`(2T)/W`
`2TW`
`(8T)/W`
`8TW`

Show Answers Only

`A`

Show Worked Solution

`(12T^2)/(3Txx2W)=(12T^2)/(6TW)=(2T)/W`

`=>\ A`

L&E, 2ADV E1 2011 HSC 1c

Solve `2^(2x+1)=32`. (2 marks)

Show Answer Only

`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`