Perform the operations and simplify7\sqrt(20-\sqrt(80+3\sqrt(45)))
First, let's simplify the expression under the square root:
\( 80 + 3\sqrt{45} \)
We know that \( 45 = 3 \times 3 \times 5 = 9 \times 5 \), so the expression becomes:
\( 80 + 3\sqrt{45} = 80 + 3\sqrt{9 \times 5} = 80 + 3 \times 3\sqrt{5} = 80 + 9\sqrt{5} \)
Now, plug this back into the original expression:
\( 7\sqrt{20 - \sqrt{80 + 3\sqrt{45}}} = 7\sqrt{20 - \sqrt{80 + 9\sqrt{5}}} \)
Now, simplify the outer square root:
\[ 7\sqrt{20 - \sqrt{80 + 9\sqrt{5}}} = 7\sqrt{20 - (80 + 9\sqrt{5})} \]
\[= 7\sqrt{20 - 80 - 9\sqrt{5}} = 7\sqrt{-60 - 9\sqrt{5}} \]
This expression cannot be simplified further because it involves a complex number and a square root within a square root.