To simplify the expression \(5\sqrt{7} - \sqrt{-8\sqrt{7}}\), we proceed as follows:
First, we note that \(\sqrt{-8\sqrt{7}} = \sqrt{-1 \cdot 8\sqrt{7}} = i\sqrt{8\sqrt{7}}\). Now we simplify \(\sqrt{8\sqrt{7}}\):
\[ \sqrt{8\sqrt{7}} = \sqrt{8} \cdot \sqrt[4]{7} \]
We know that \(\sqrt{8} = 2\sqrt{2}\). Thus:
\[ \sqrt{8\sqrt{7}} = 2\sqrt{2} \cdot \sqrt[4]{7} \]
This gives us:
\[ \sqrt{-8\sqrt{7}} = i \cdot 2\sqrt{2} \cdot \sqrt[4]{7} \]
Now substituting back into the expression:
\[ 5\sqrt{7} - i \cdot 2\sqrt{2} \cdot \sqrt[4]{7} \]
Since the expression has a complex component with \(i\), and the other terms are real and involve \(\sqrt{7}\), we can conclude that the expression has both real and imaginary parts. However, if we consider the context of the simplification you suggested, it appears to ask for simplification without complex numbers unless required.
However, without knowing the context (i.e., whether complex numbers should be included), we will retain our previous calculation.
To choose among the options provided:
- -3√7
- -3√14
- 13√7
- -21x
None of these given options match the original expression correctly when simplified under conventional methods.
If we misinterpreted the terms or the nature of the terms in your problem, please provide a clarification or revision of the expression, especially related to the square root of negative values, as it is crucial to understand whether you want a real-valued expression or if complex numbers are permissible.