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# sqrt(5)+sqrt(3)) / (sqrt(5)-sqrt(3)) =

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Manager
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13 Sep 2005, 13:36
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$$\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} =$$
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13 Sep 2005, 14:14
(sqrt(5)+sqrt(3)) / (sqrt(5)-sqrt(3))

multiple both numerator and denominator by :
(sqrt(5)+sqrt(3))

gives:

{[(sqrt(5)+sqrt(3))] ^ 2} / 2

or

4 + sqrt (15)
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13 Sep 2005, 20:34
the reason i posted this question was to ask a second question.

if you chose to multiply the numerator and denominator by sqrt(5)+sqrt(3) you get 4+sqrt(5)*sqrt(3)

though, if you chose to multiply the numerator and denominator by sqrt(5)-sqrt(3) you get 1 / (4-sqrt(5)*sqrt(3))

though seemingly different,

4+sqrt(5)*sqrt(3) = 1/(4-sqrt(5)*sqrt(3))

I am not sure whether there is a broad generalization i am missing. If anyone can shed light on this, that would be great!
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13 Sep 2005, 23:56
chets wrote:
(sqrt(5)+sqrt(3)) / (sqrt(5)-sqrt(3)) = ?

Take conjugate on both sides. which gives.

(sqrt(5) + sqrt(3)) (sqrt(5) + sqrt(3)) / (sqrt(5) - sqrt(3)) (sqrt(5) + sqrt(3))

This gives (sqrt(5) +sqrt(3))^2 / (Sqrt(5)^2) - (sqrt(3))^2

=> 5 + 3 + 2 sqrt(15) / 2

=> 2(4 + sqrt(15))/2

=> 4 + sqrt(15)

Thanks
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14 Sep 2005, 06:34
chets wrote:
the reason i posted this question was to ask a second question.

if you chose to multiply the numerator and denominator by sqrt(5)+sqrt(3) you get 4+sqrt(5)*sqrt(3)

though, if you chose to multiply the numerator and denominator by sqrt(5)-sqrt(3) you get 1 / (4-sqrt(5)*sqrt(3))

though seemingly different,

4+sqrt(5)*sqrt(3) = 1/(4-sqrt(5)*sqrt(3))

I am not sure whether there is a broad generalization i am missing. If anyone can shed light on this, that would be great!

Chet,

You r not missing anything,

You get:

1/(4-sqrt(5)*sqrt(3)) =

(4+sqrt(5)*sqrt(3)) / [(4-sqrt(5)*sqrt(3))(4+sqrt(5)*sqrt(3))] =

(4+sqrt(5)*sqrt(3)) / (16 - 15) =

(4+sqrt(5)*sqrt(3))

Hope that helps
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Re: sqrt(5)+sqrt(3)) / (sqrt(5)-sqrt(3)) = [#permalink]

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21 May 2014, 08:38
This type of questions can be solved by using formula

(a+b)(a-b) = a^2 - b^2

Denominator is in the form of a-b , hence we will multiply numerator and denominator by the form a +b .i.e. sqrt(5) + sqrt(3)

sqrt(5) + sqrt(3)/sqrt(5) - sqrt(3)

[sqrt(5) + sqrt(3)]*[sqrt(5) + sqrt(3)]/[sqrt(5) + sqrt(3)]*[sqrt(5) - sqrt(3)]

5+3 + 2*sqrt(5)*sqrt(3) / 5-3

8+2*sqrt(5)*sqrt(3)/2

4+sqrt(15)
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Re: sqrt(5)+sqrt(3)) / (sqrt(5)-sqrt(3)) = [#permalink]

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22 May 2014, 03:33
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Re: sqrt(5)+sqrt(3)) / (sqrt(5)-sqrt(3)) = [#permalink]

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22 May 2014, 14:59
Chets asked if 4+sqrt(5)*sqrt(3) = 1/(4-sqrt(5)*sqrt(3)) can be extrapolated into a broad generalization. No, it can't be!

Here's why.

Consider an expression $$1/(a-\sqrt{bc})$$ [modeled on 1/(4-sqrt(5)*sqrt(3)) ]
Multiplying Numerator and Denominator with $$(a+ \sqrt{bc})$$, we get
$$(a+ \sqrt{bc})/(a^2-bc)$$

So, the reason why 1/(4-sqrt(5)*sqrt(3) got simplified into 4+sqrt(5)*sqrt(3) was because the Denominator of the above expression is equal to 1 in this particular case (a=4, b=5, c=3. So,$$a^2-bc = 4^2-5*3 = 1$$)
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Re: sqrt(5)+sqrt(3)) / (sqrt(5)-sqrt(3)) = [#permalink]

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22 May 2014, 19:37
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Multiply numerator & denominator by $$(\sqrt{5} + \sqrt{3})$$

$$= \frac{(\sqrt{5} + \sqrt{3})^2}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})}$$

$$= \frac{5 + 2\sqrt{15} + 3}{5 - 3}$$

$$= \frac{8 + 2\sqrt{15}}{2}$$

$$= 4 + \sqrt{15}$$

Formulae used:

$$(a+b)^2 = a^2 + 2ab + b^2$$

$$(a-b)^2 = a^2 - b^2$$

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Re: sqrt(5)+sqrt(3)) / (sqrt(5)-sqrt(3)) = [#permalink]

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09 Jul 2015, 07:53
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Re: sqrt(5)+sqrt(3)) / (sqrt(5)-sqrt(3)) =   [#permalink] 09 Jul 2015, 07:53
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