smc-6213-30-Surd Denominators

Functions, 2ADV EQ-Bank 03

Find \(a\) and \(b\) such that \(a\) and \(b\) are real and \(\dfrac{2\sqrt{3}+2}{\sqrt{6}-\sqrt{2}} = a\,\sqrt{2} + b\,\sqrt{6}\). (2 marks)

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\(a=2\ \ \text{and}\ \ b=1 \)

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\(\dfrac{2\sqrt{3}+2}{\sqrt{6}-\sqrt{2}} \times \dfrac{\sqrt{6}+\sqrt{2}}{\sqrt{6}+\sqrt{2}} \)

\(=\dfrac{(2\sqrt{3}+2)(\sqrt{6}+\sqrt{2})}{6-2}\)

\(=\dfrac{2\sqrt{18}+2\sqrt{6}+2\sqrt{6}+2\sqrt{2}}{4}\)

\(=\dfrac{6\sqrt{2}+4\sqrt{6}+2\sqrt{2}}{4}\)

\(=\dfrac{8\sqrt{2}+4\sqrt{6}}{4}\)

\(=2\sqrt{2}+\sqrt{6}\)

\(\therefore a=2\ \ \text{and}\ \ b=1 \)

Functions, 2ADV EQ-Bank 02

Rationalise the denominator in \(\dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}\), and express in the simplest form. (2 marks)

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\(\dfrac{\sqrt{15}-\sqrt{6}-\sqrt{10}+2}{3} \)

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\(\dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{5}+\sqrt{2}} \times \dfrac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}}\)

\(=\dfrac{(\sqrt{3}-\sqrt{2})(\sqrt{5}-\sqrt{2})}{5-2}\)

\(=\dfrac{\sqrt{15}-\sqrt{6}-\sqrt{10}+2}{3}\)

Functions, 2ADV EQ-Bank 01

Find \(x\) and \(y\) such that \(x\) and \(y\) are real and \(\dfrac{\sqrt{2}+1}{\sqrt{6}-\sqrt{3}} = x\,\sqrt{3} + y\,\sqrt{6}\). (2 marks)

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\(x=1\ \ \text{and}\ \ y=\dfrac{2}{3} \)

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\(\dfrac{\sqrt{2}+1}{\sqrt{6}-\sqrt{3}} \times \dfrac{\sqrt{6}+\sqrt{3}}{\sqrt{6}+\sqrt{3}} \)

\(=\dfrac{(\sqrt{2}+1)(\sqrt{6}+\sqrt{3})}{6-3}\)

\(=\dfrac{\sqrt{12}+2\sqrt{6}+\sqrt{3}}{3}\)

\(=\dfrac{3\sqrt{3}+2\sqrt{6}}{3}\)

\(= \sqrt{3}+\dfrac{2}{3} \sqrt{6}\)

\(\therefore x=1\ \ \text{and}\ \ y=\dfrac{2}{3} \)

Functions, 2ADV F1 SM-Bank 52

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`

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`(6sqrt3-sqrt5)/(2sqrt5)`	`=(6sqrt3-sqrt5)/(2sqrt5) xx (2sqrt5)/(2sqrt5)`
	`=(2sqrt5(6sqrt3 – sqrt5))/(4 xx5)`
	`=(12sqrt15-10)/20`
	`=- 1/2 + 3/5 sqrt15`

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

Functions, 2ADV F1 SM-Bank 51

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

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

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

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`a = 1/2, \ b = – 1/3`

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`(sqrt3-2)/(2sqrt3)`	`= (sqrt3-2)/(2sqrt3) xx (2sqrt3)/(2sqrt3)`
	`= (2sqrt3(sqrt3-2))/(4 xx 3)`
	`= (6-4sqrt3)/12`
	`=1/2 – 1/3 sqrt3`

`:.\ a = 1/2, \ b = – 1/3`

Functions, 2ADV F1 SM-Bank 46

Find `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`

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`(8-sqrt27)/(2sqrt3) xx (2sqrt3)/(2sqrt3)`	`=(2sqrt3(8-3sqrt3))/(2sqrt3)^2`
	`= (16sqrt3-18)/12`
	`= -3/2 + 4/3sqrt3`

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

Functions, 2ADV F1 2017 HSC 11a

Rationalise the denominator of `2/(sqrt(5)-1)`. (2 marks)

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`(sqrt(5) + 1)/2`

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`2/(sqrt(5)-1) xx (sqrt(5) + 1)/(sqrt(5) + 1)`	`= (2(sqrt(5) + 1))/((sqrt 5)^2-1)`
	`= (2(sqrt(5) + 1))/4`
	`= (sqrt(5) + 1)/2`

Functions, 2ADV F1 2015 HSC 11c

Express `8/(2 + sqrt 7)` with a rational denominator. (2 marks)

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`(-8 (2-sqrt 7))/3`

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`8/(2 + sqrt 7) xx (2-sqrt 7)/(2-sqrt 7)`

`= (8(2-sqrt 7))/(2^2 -(sqrt 7)^2)`

`= (8 (2-sqrt 7))/(4-7)`

`= (-8(2-sqrt 7))/3`

Functions, 2ADV F1 2014 HSC 11a

Rationalise the denominator of `1/(sqrt5-2)`. (2 marks)

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`sqrt5 + 2`

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`1/(sqrt5-2) xx (sqrt5 + 2)/(sqrt5 + 2)`

`= (sqrt5 + 2)/((sqrt5)^2-2^2)`

`= sqrt5 + 2`

Functions, 2ADV F1 2011 HSC 1f

Rationalise the denominator of `4/(sqrt5-sqrt3)`.

Give your answer in the simplest form. (2 marks)

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`2(sqrt5 + sqrt3)`

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`4/(sqrt5-sqrt3) xx (sqrt5 + sqrt3)/(sqrt5 + sqrt3)`

`= (4(sqrt5 + sqrt3))/(5-3)`

`= 2 (sqrt5 + sqrt3)`

Functions, 2ADV F1 2012 HSC 2 MC

Which of the following is equal to `1/(2sqrt5-sqrt3)`?

`(2sqrt5\-sqrt3)/7`
`(2sqrt5 + sqrt3)/7`
`(2sqrt5\-sqrt3)/17`
`(2sqrt5 + sqrt3)/17`

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`D`

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`1/(2sqrt5\-sqrt3) xx (2sqrt5 + sqrt3)/(2sqrt5 + sqrt3)`

`= (2sqrt5 + sqrt3)/( (2sqrt5\-sqrt3)(2sqrt5 + sqrt3) )`

`= (2sqrt5 + sqrt3)/{(2sqrt5)^2-(sqrt3)^2)`

`= (2sqrt5 + sqrt3)/17`

`=> D`