Solve for: sqrt(7+4\sqrt{3)}+sqrt(7-4\sqrt{3)}

Expression: $\sqrt{ 7+4\sqrt{ 3 } }+\sqrt{ 7-4\sqrt{ 3 } }$

Use ${a}^{2}+2ab+{b}^{2}={\left( a+b \right)}^{2}$ to factor the expression

$\sqrt{ {\left( 2+\sqrt{ 3 } \right)}^{2} }+\sqrt{ 7-4\sqrt{ 3 } }$

Use ${a}^{2}-2ab+{b}^{2}={\left( a-b \right)}^{2}$ to factor the expression

$\sqrt{ {\left( 2+\sqrt{ 3 } \right)}^{2} }+\sqrt{ {\left( 2-\sqrt{ 3 } \right)}^{2} }$

Reduce the index of the radical and exponent with $2$

$2+\sqrt{ 3 }+\sqrt{ {\left( 2-\sqrt{ 3 } \right)}^{2} }$

Reduce the index of the radical and exponent with $2$

$2+\sqrt{ 3 }+2-\sqrt{ 3 }$

Since two opposites add up to $0$, remove them from the expression

$2+2$

Add the numbers

$4$