# Solve for: sqrt(3)sqrt(6+\sqrt{3)}sqrt(3+\sqrt{3+\sqrt{3)}}sqrt(3-\sqrt{3+\sqrt{3)}}

## Expression: $\sqrt{ 3 }\sqrt{ 6+\sqrt{ 3 } }\sqrt{ 3+\sqrt{ 3+\sqrt{ 3 } } }\sqrt{ 3-\sqrt{ 3+\sqrt{ 3 } } }$

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

$\sqrt{ 3\left( 6+\sqrt{ 3 } \right) \times \left( 3+\sqrt{ 3+\sqrt{ 3 } } \right) \times \left( 3-\sqrt{ 3+\sqrt{ 3 } } \right) }$

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

$\sqrt{ 3\left( 6+\sqrt{ 3 } \right) \times \left( 9-\left( 3+\sqrt{ 3 } \right) \right) }$

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

$\sqrt{ 3\left( 6+\sqrt{ 3 } \right) \times \left( 9-3-\sqrt{ 3 } \right) }$

Subtract the numbers

$\sqrt{ 3\left( 6+\sqrt{ 3 } \right) \times \left( 6-\sqrt{ 3 } \right) }$

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

$\sqrt{ 3\left( 36-3 \right) }$

Subtract the numbers

$\sqrt{ 3 \times 33 }$

Multiply the numbers

$\sqrt{ 99 }$

\begin{align*}&3\sqrt{ 11 } \\&\approx9.94987\end{align*}