Rationalizing radical denominators

errequeerre

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Jun 18, 2013
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Hi,

I am doing an exercise to practice rationalizing denominators, and I have come across this denominator: (-3+~5)

I was in doubt whether the conjugate was (3-~5) or (-~5+3).

Apparently it is neither, and I am told that the conjugate is (-3-~5).

I can't understand why. Could you help me?
 
I am doing an exercise to practice rationalizing denominators, and I have come across this denominator: (-3+~5)
I was in doubt whether the conjugate was (3-~5) or (-~5+3).
Apparently it is neither, and I am told that the conjugate is (-3-~5).
I can't understand why. Could you help me?
I see that you are in Spain. I tell you that I have never seen the notation (-3-~5).

But you said the conjugate and rationalize.

Note that \(\displaystyle \left( {3 + \sqrt 5 } \right)\left( {3 - \sqrt 5 } \right) = 4\) which is rational.

So \(\displaystyle \dfrac{1}{{\left( {3 + \sqrt 5 } \right)}} = \dfrac{{\left( {3 - \sqrt 5 } \right)}}{4}\)
 
Hi pka,

Thanks for your post. I'm afraid, though, that I can't still understand this.

I've learnt that the conjugate is the expression with opposite signs, and I understand that it helps you get rid of the radical expression in the denominator.

My problem is that the original denominator is -3 + ~5 (I read that the ~-sign can be used instead of the root sign when you can't type it), which could also be stated as ~5 - 3.

I think of two equivalent possibilities to change the signs of the expression above:
3 - ~5 or - ~5 + 3
However, the book I'm learning with says that the conjugate is -3 - ~5, and I cannot understand why, since - ~5 - 3 is not the same as - ~5 + 3.

Thanks for your help, anyway
 
Hi pka,

Thanks for your post. I'm afraid, though, that I can't still understand this.

I've learnt that the conjugate is the expression with opposite signs, and I understand that it helps you get rid of the radical expression in the denominator.

My problem is that the original denominator is -3 + ~5 (I read that the ~-sign can be used instead of the root sign when you can't type it), which could also be stated as ~5 - 3.

I think of two equivalent possibilities to change the signs of the expression above:
3 - ~5 or - ~5 + 3
However, the book I'm learning with says that the conjugate is -3 - ~5, and I cannot understand why, since - ~5 - 3 is not the same as - ~5 + 3.

Thanks for your help, anyway

conjugate of (√5 - 3) is (-√5 - 3)

Only the number with √ changes sign. This comes from the fact:

(a + b) (a - b) = a2 - b2
 
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