+0

sqrt(8-sqrt(55))+sqrt(8+sqrt(55))

0
330
5

sqrt(8-sqrt(55))+sqrt(8+sqrt(55))

Guest Jan 25, 2015

#3
+18834
+10

sqrt(8-sqrt(55))+sqrt(8+sqrt(55))

$$\sqrt{8-\sqrt{55}}+\sqrt{8+\sqrt{55}} \quad | \quad \sqrt{ x^2 } \\ \\ = \sqrt{ \left( \sqrt{8-\sqrt{55}}+\sqrt{8+\sqrt{55}} \right)^2 } \\ \\ = \sqrt{ (8-\sqrt{55}) +(8+ \sqrt{55}) +2*\left( \sqrt{8-\sqrt{55}} \right)* \left(\sqrt{8+\sqrt{55}} \right)} \\ \\ = \sqrt{ 16+2*\left( \sqrt{8-\sqrt{55}} \right) * \left(\sqrt{8+\sqrt{55}} \right) } \\ \\ = \sqrt{ 16+2* \sqrt{8^2- 55 } }\\ \\ = \sqrt{ 16+2* \sqrt{64-55} } \\ \\ = \sqrt{ 16+2* \sqrt{9} } \\ \\ = \sqrt{ 16+2* 3 }\\ \\ = \sqrt{ 22 }\\ \\ = 4.69041576$$

heureka  Jan 25, 2015
Sort:

#1
+91479
+10

$$\sqrt{8-\sqrt{55}}+\sqrt{8+\sqrt{55}}$$

This is a tough one!

First I am going to consider      $$8-\sqrt{55}$$

I want to express this as a perfect square.

$$\\8-\sqrt{55}\\\\ =\frac{5}{2}-\sqrt{55}+\frac{11}{2}\\\\ =\frac{25}{10}-\frac{10\sqrt{55}}{10}+\frac{55}{10}\\\\ =\frac{25-10\sqrt{55}+55}{10}\\\\ =\frac{5^2-10\sqrt{55}+(\sqrt{55})^2}{10}\\\\ =\frac{(5-\sqrt{55})^2}{10}\\\\ But\;\; (5-\sqrt{55})^2=(\sqrt{55}-5)^2\;\;and I want the positive one, so \\\\ =\frac{(\sqrt{55}-5)^2}{10}\\\\$$

$$\\Hence\\\\ \sqrt{8-\sqrt{55}}\\\\ =\sqrt{\frac{(\sqrt{55}-5)^2}{10}}\\\\ =\frac{\sqrt{55}-5}{\sqrt{10}}}\\\\ =\frac{\sqrt{10*55}-5\sqrt{10}}{10}\\\\ =\frac{\sqrt{2*5*5*11}-5\sqrt{10}}{10}\\\\ =\frac{5\sqrt{22}-5\sqrt{10}}{10}\\\\ =\frac{\sqrt{22}-\sqrt{10}}{2}\\\\$$

NOW, BY THE SAME LOGIC,

$$\\Hence\\\\ \sqrt{8}+\sqrt{55}=\frac{\sqrt{22}+\sqrt{10}}{2}\\\\ SO\\\\ \sqrt{8-\sqrt{55}}+\sqrt{8+\sqrt{55}}\\\\ =\frac{\sqrt{22}-\sqrt{10}}{2}+\frac{\sqrt{22}+\sqrt{10}}{2}\\\\ =\frac{2\sqrt{22}}{2}\\\\ =\sqrt{22}\\\\$$

Melody  Jan 25, 2015
#2
+81068
0

Very crafty, Melody....!!!

CPhill  Jan 25, 2015
#3
+18834
+10

sqrt(8-sqrt(55))+sqrt(8+sqrt(55))

$$\sqrt{8-\sqrt{55}}+\sqrt{8+\sqrt{55}} \quad | \quad \sqrt{ x^2 } \\ \\ = \sqrt{ \left( \sqrt{8-\sqrt{55}}+\sqrt{8+\sqrt{55}} \right)^2 } \\ \\ = \sqrt{ (8-\sqrt{55}) +(8+ \sqrt{55}) +2*\left( \sqrt{8-\sqrt{55}} \right)* \left(\sqrt{8+\sqrt{55}} \right)} \\ \\ = \sqrt{ 16+2*\left( \sqrt{8-\sqrt{55}} \right) * \left(\sqrt{8+\sqrt{55}} \right) } \\ \\ = \sqrt{ 16+2* \sqrt{8^2- 55 } }\\ \\ = \sqrt{ 16+2* \sqrt{64-55} } \\ \\ = \sqrt{ 16+2* \sqrt{9} } \\ \\ = \sqrt{ 16+2* 3 }\\ \\ = \sqrt{ 22 }\\ \\ = 4.69041576$$

heureka  Jan 25, 2015
#4
+81068
0

Also well done, heureka...!!!

Your method is a little more intuitive to me, than Melody's.....

But.......either one gets the job done!!!

CPhill  Jan 25, 2015
#5
+91479
0

Thanks Chris,

Yes, I will admit, I like Heureka's method better too.

Melody  Jan 25, 2015

23 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details