Rationalizing the denominator: (3 + √2) / (4 - √2), 1 / 1-√5

[Banks]

Could someone clarify this please.

I've been asked to simplify the following:

1. (3 + √2) / (4 - √2)

2. 1 / 1-√5

For #1, I got this answer: 1 + 7√2 but in the book, it says the correct answer is: 1 + ½√2
For #2, I got this answer: - ¼ + √5 but the books says it's: - ¼ (1 + √5)

I don't understand how either came to those answers, could someone please explain? I especially don't understand #2.

Thanks in advance

 

[MarkFL]

1.) \(\displaystyle \displaystyle \frac{3+\sqrt{2}}{4-\sqrt{2}}\cdot\frac{4+\sqrt{2}}{4+\sqrt{2}}=\frac{12+7\sqrt{2}+2}{16-2}=\frac{7(2+\sqrt{2})}{14}=\frac{2+\sqrt{2}}{2}=1+\frac{1}{2}\sqrt{2}\)

2.) \(\displaystyle \displaystyle \frac{1}{1-\sqrt{5}}=-\frac{1}{\sqrt{5}-1}\cdot\frac{\sqrt{5}+1}{\sqrt{5}+1}=-\frac{\sqrt{5}+1}{5-1}=-\frac{\sqrt{5}+1}{4}=-\frac{1}{4}(1+\sqrt{5})\)

 

[Banks]

I see how #1 came to be that now. For #2 though, could you please explain how (1 / 1 - √5) ...became negative... - (1 / √5 - 1) and why you switched the surd and 1 around in the denominator? Sorry I don't understand that bit.

Thanks for your help.

 

[JeffM]

I see how #1 came to be that now. For #2 though, could you please explain how (1 / 1 - √5) ...became negative... - (1 / √5 - 1) and why you switched the surd and 1 around in the denominator? Sorry I don't understand that bit.

Thanks for your help.

\(\displaystyle \dfrac{1}{1 - \sqrt{5}} = \dfrac{1}{1 - \sqrt{5}} * 1 = \dfrac{1}{1 - \sqrt{5}} * \dfrac{1 + \sqrt{5}}{1 + \sqrt{5}} =\)

\(\displaystyle \dfrac{1 + \sqrt{5}}{(1 - \sqrt{5})(1 + \sqrt{5})} = \dfrac{1 + \sqrt{5}}{1 - (\sqrt{5})^2} = \dfrac{1 + \sqrt{5}}{1 - 5} =\)

\(\displaystyle \dfrac{1 + \sqrt{5}}{-\ 4} = -\ \dfrac{1}{4} * (1 + \sqrt{5}).\)

 

[MarkFL]

I see how #1 came to be that now. For #2 though, could you please explain how (1 / 1 - √5) ...became negative... - (1 / √5 - 1) and why you switched the surd and 1 around in the denominator? Sorry I don't understand that bit.

Thanks for your help.

Consider:

\(\displaystyle \displaystyle \frac{1}{a-b}=\frac{1}{-(b-a)}=-\frac{1}{b-a}\)

As Jeff showed, this is not a necessary step, but knowing \(\displaystyle 1-\sqrt{5}<0\), I wanted to pull that negative sign out front immediately, and then work with a positive denominator.