Rationalising problem

Algebra Level 2

$\large\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}-\sqrt{3-2\sqrt{2}} = \, ?$

1 solution

Brian Charlesworth
Jan 6, 2016

Note first that $(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2})^{2} =$

$(\sqrt{5} + 2) + (\sqrt{5} - 2) + 2\sqrt{(\sqrt{5} + 2)(\sqrt{5} - 2)} = 2\sqrt{5} + 2\sqrt{5 - 4} = 2(\sqrt{5} + 1).$

Next note that $3 - 2\sqrt{2} = 2 - 2\sqrt{2} + 1 = (\sqrt{2})^{2} - 2\sqrt{2} + 1 = (\sqrt{2} - 1)^{2}$ .

The given expression can then be written as

$\dfrac{\sqrt{2(\sqrt{5} + 1)}}{\sqrt{\sqrt{5} + 1}} - \sqrt{(\sqrt{2} - 1)^{2}} = \sqrt{2} - (\sqrt{2} - 1) = \boxed{1}.$

Nice solution Brian:)

Racchit Jain - 5 years, 5 months ago

Where'd you get that question? ASAT?

Aditya Agarwal - 5 years, 5 months ago

Nope it was given to me by my cousin. I guess he goes to ALLEN so it must be from his maths module.

Racchit Jain - 5 years, 5 months ago

Thanks! Nice problem. :)

Brian Charlesworth - 5 years, 5 months ago

@Racchit Jain , i remembered this question is copied from SOF imo workbook class 9 ch-1 q.6 :-)

sanyam goel - 4 years, 11 months ago

Idk, maybe, coaching institutes copy problems from books, so I guess so :)

Racchit Jain - 4 years, 11 months ago