roots problem

Simplify the following:
2/sqrt(4 sqrt(2)+6)+2/sqrt(6-4 sqrt(2))

Factor 2 out of 6+4 sqrt(2) giving 2 (3+2 sqrt(2)):
2/( sqrt(2 (3+2 sqrt(2)) ) )+2/sqrt(6-4 sqrt(2))

Factor 2 out of 6-4 sqrt(2) giving 2 (3-2 sqrt(2)):
2/sqrt(2 (3+2 sqrt(2)))+2/( sqrt(2 (3-2 sqrt(2)) ) )

2 (3+2 sqrt(2))  =  4+4 sqrt(2)+2  =  4+4 sqrt(2)+(sqrt(2))^2  =  (2+sqrt(2))^2:
2/( sqrt((2+sqrt(2))^2 ) )+2/sqrt(2 (3-2 sqrt(2)))

Cancel exponents. sqrt((2+sqrt(2))^2) = 2+sqrt(2):
2/2+sqrt(2)+2/sqrt(2 (3-2 sqrt(2)))

2 (3-2 sqrt(2))  =  4-4 sqrt(2)+2  =  4-4 sqrt(2)+(sqrt(2))^2  =  (2-sqrt(2))^2:
2/(2+sqrt(2))+2/( sqrt((2-sqrt(2))^2 ) )

Cancel exponents. sqrt((2-sqrt(2))^2) = 2-sqrt(2):
2/(2+sqrt(2))+2/2-sqrt(2)

Multiply numerator and denominator of 2/(2+sqrt(2)) by 2-sqrt(2):
(2 (2-sqrt(2)))/((2+sqrt(2)) (2-sqrt(2)))+2/(2-sqrt(2))

(2+sqrt(2)) (2-sqrt(2)) = 2×2+2 (-sqrt(2))+sqrt(2)×2+sqrt(2) (-sqrt(2)) = 4-2 sqrt(2)+2 sqrt(2)-2 = 2:
(2 (2-sqrt(2)))/(2)+2/(2-sqrt(2))

(2 (2-sqrt(2)))/(2) = 2/2×(2-sqrt(2)) = 2-sqrt(2):
2-sqrt(2)+2/(2-sqrt(2))

Multiply numerator and denominator of 2/(2-sqrt(2)) by 2+sqrt(2):
2-sqrt(2)+(2 (2+sqrt(2)))/((2-sqrt(2)) (2+sqrt(2)))

(2-sqrt(2)) (2+sqrt(2)) = 2×2+2 sqrt(2)-sqrt(2)×2-sqrt(2) sqrt(2) = 4+2 sqrt(2)-2 sqrt(2)-2 = 2:
2-sqrt(2)+(2 (2+sqrt(2)))/(2)

(2 (2+sqrt(2)))/(2) = 2/2×(2+sqrt(2)) = 2+sqrt(2):
2-sqrt(2)+2+sqrt(2)

2-sqrt(2)+2+sqrt(2) = 4:
Answer: |  4

how do I solve 2/(sqrt(4sqrt(2)+6))+2/(sqrt(6-4*sqrt(2)) , without calculator, it's been an hour at least..

\(\begin{array}{rcll} \frac{2} {\sqrt{ 6 + 4\cdot \sqrt{2}} } + \frac{2} { \sqrt{6-4 \cdot \sqrt{2} } } &=& 2\cdot \left( \frac{1} {\sqrt{ 6 + 4\cdot \sqrt{2}} } + \frac{1} { \sqrt{6-4 \cdot \sqrt{2} } } \right) \\\\ &=& 2\cdot \left( \frac{ \sqrt{ 6 - 4 \cdot \sqrt{2} } + \sqrt{6+4 \cdot \sqrt{2} } } { \sqrt{ 6 + 4 \cdot \sqrt{2} } \cdot \sqrt{ 6 - 4 \cdot \sqrt{2} } } \right) \\\\ &=& 2\cdot \left( \frac{ \sqrt{ 6 - 4 \cdot \sqrt{2} } + \sqrt{6+4 \cdot \sqrt{2} } } { \sqrt{ ( 6 + 4 \cdot \sqrt{2} ) \cdot ( 6 - 4 \cdot \sqrt{2} ) } } \right) \\\\ &=& 2\cdot \left( \frac{ \sqrt{ 6 - 4 \cdot \sqrt{2} } + \sqrt{6+4 \cdot \sqrt{2} } } { \sqrt{ 36 - 16 \cdot 2 } } \right) \\\\ &=& 2\cdot \left( \frac{ \sqrt{ 6 - 4 \cdot \sqrt{2} } + \sqrt{6+4 \cdot \sqrt{2} } } { \sqrt{ 4 } } \right) \\\\ &=& 2\cdot \left( \frac{ \sqrt{ 6 - 4 \cdot \sqrt{2} } + \sqrt{6+4 \cdot \sqrt{2} } } { 2 } \right) \\\\ &=& \sqrt{ 6 - 4 \cdot \sqrt{2} } + \sqrt{6+4 \cdot \sqrt{2} } \\\\ \end{array}\)

\(\begin{array}{rcll} \frac{2} {\sqrt{ 6 + 4\cdot \sqrt{2}} } + \frac{2} { \sqrt{6-4 \cdot \sqrt{2} } } &=& \sqrt{ 6 - 4 \cdot \sqrt{2} } + \sqrt{6+4 \cdot \sqrt{2} } \qquad | \qquad \sqrt{()^2} \\\\ &=& \sqrt{ \left( \sqrt{ 6 - 4 \cdot \sqrt{2} } + \sqrt{6+4 \cdot \sqrt{2} } \right)^2 } \\\\ &=& \sqrt{ (6-4 \cdot \sqrt{2}) + 2\cdot \sqrt{ 6 - 4 \cdot \sqrt{2} }\cdot \sqrt{6+4 \cdot \sqrt{2} } + (6+4 \cdot \sqrt{2}) } \\\\ &=& \sqrt{ 6 + 2\cdot \sqrt{ 6 - 4 \cdot \sqrt{2} }\cdot \sqrt{6+4 \cdot \sqrt{2} } + 6 } \\\\ &=& \sqrt{ 12 + 2\cdot \sqrt{ 6 - 4 \cdot \sqrt{2} }\cdot \sqrt{6+4 \cdot \sqrt{2} } } \\\\ &=& \sqrt{ 12 + 2\cdot \sqrt{ (6 - 4 \cdot \sqrt{2}) \cdot (6+4 \cdot \sqrt{2}) } } \\\\ &=& \sqrt{ 12 + 2\cdot \sqrt{ 36 - 16 \cdot 2 } } \\\\ &=& \sqrt{ 12 + 2\cdot \sqrt{ 4 } } \\\\ &=& \sqrt{ 12 + 2\cdot 2 } \\\\ &=& \sqrt{ 12 + 4 } \\\\ &=& \sqrt{ 16 } \\\\ \mathbf{ \frac{2} {\sqrt{ 6 + 4\cdot \sqrt{2}} } + \frac{2} { \sqrt{6-4 \cdot \sqrt{2} } } } &\mathbf{=}& \mathbf{4} \end{array}\)