# Solve for: sqrt(2+\sqrt{3)}sqrt(2+\sqrt{2+\sqrt{3)}}sqrt(2-\sqrt{2+\sqrt{3)}}

## Expression: $\sqrt{ 2+\sqrt{ 3 } }\sqrt{ 2+\sqrt{ 2+\sqrt{ 3 } } }\sqrt{ 2-\sqrt{ 2+\sqrt{ 3 } } }$

The product of roots with the same index is equal to the root of the product

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

Use $\left( a-b \right)\left( a+b \right)={a}^{2}-{b}^{2}$ to simplify the product

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

When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses

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

Subtract the numbers

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

Use $\left( a-b \right)\left( a+b \right)={a}^{2}-{b}^{2}$ to simplify the product

$\sqrt{ 4-3 }$

Subtract the numbers

$\sqrt{ 1 }$

Any root of $1$ equals $1$

$1$

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