Question

4÷ whole root of 6+ √20- 2÷ whole root of 8+ √60- 2÷ whole root of 4-√12=

Answer 1

SOLUTION

TO SIMPLIFY

$\displaystyle \sf{ \frac{4}{ \sqrt{6 + \sqrt{20} }} - \frac{2}{ \sqrt{ 8 + \sqrt{60} }} - \frac{2}{ \sqrt{ 4 - \sqrt{12} } }}$

EVALUATION

$\displaystyle \sf{ \frac{4}{ \sqrt{6 + \sqrt{20} }} - \frac{2}{ \sqrt{ 8 + \sqrt{60} }} - \frac{2}{ \sqrt{ 4 - \sqrt{12} } }}$

$\displaystyle \sf{ = \frac{4}{ \sqrt{6 + \sqrt{4 \times 5} }} - \frac{2}{ \sqrt{ 8 + \sqrt{4 \times 15} }} - \frac{2}{ \sqrt{ 4 - \sqrt{4 \times 3} } }}$

$\displaystyle \sf{ = \frac{4}{ \sqrt{6 + 2\sqrt{ 5} }} - \frac{2}{ \sqrt{ 8 + 2\sqrt{ 15} }} - \frac{2}{ \sqrt{ 4 - 2\sqrt{ 3} } }}$

$\displaystyle \sf{ = \frac{4}{ \sqrt{5 + 1 + 2\sqrt{ 5} }} - \frac{2}{ \sqrt{ 5 + 3 + 2\sqrt{ 15} }} - \frac{2}{ \sqrt{ 3 + 1 - 2\sqrt{ 3} } }}$

$\displaystyle \sf{ = \frac{4}{ \sqrt{ {( \sqrt{5} + 1)}^{2} }} - \frac{2}{ \sqrt{ {( \sqrt{5} + \sqrt{3} )}^{2} }} - \frac{2}{ \sqrt{ {( \sqrt{3} - 1)}^{2} } }}$

$\displaystyle \sf{ = \frac{4}{ ( \sqrt{5} + 1)} - \frac{2}{ {( \sqrt{5} + \sqrt{3} ) }} - \frac{2}{ {( \sqrt{3} - 1) } }}$

$\displaystyle \sf{ = \frac{4( \sqrt{5} - 1)}{ ( \sqrt{5} + 1)( \sqrt{5} - 1)} - \frac{2( \sqrt{5} - \sqrt{3} )}{ {( \sqrt{5} + \sqrt{3} )( \sqrt{5} - \sqrt{3} ) }} - \frac{2( \sqrt{3} + 1 )}{ {( \sqrt{3} - 1)( \sqrt{3} + 1 ) } }}$

$\displaystyle \sf{ = \frac{4( \sqrt{5} - 1)}{ ( 5 - 1)} - \frac{2( \sqrt{5} - \sqrt{3} )}{ {( 5 - 3) }} - \frac{2( \sqrt{3} + 1 )}{ {(3 - 1) } }}$

$\displaystyle \sf{ = \frac{4( \sqrt{5} - 1)}{ 4} - \frac{2( \sqrt{5} - \sqrt{3} )}{ {2 }} - \frac{2( \sqrt{3} + 1 )}{ {2 } }}$

$\displaystyle \sf{ = ( \sqrt{5} - 1) - ( \sqrt{5} - \sqrt{3} ) - ( \sqrt{3} + 1 )}$

$\displaystyle \sf{ = \sqrt{5} - 1 - \sqrt{5} + \sqrt{3} - \sqrt{3} - 1 }$

$\displaystyle \sf{ = - 2}$

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