To simplify the expression \( 5\sqrt{7} - \sqrt{-8\sqrt{7}} \), we first need to clarify the expression, as it seems potentially miswritten due to the negative under a square root.
If we take \( \sqrt{-8\sqrt{7}} \) to mean taking the square root of a negative number, we can rewrite it in terms of \( i \) (the imaginary unit). However, if we look at \( -8\sqrt{7} \) with just the numeric portion, we can identify its components.
We proceed as follows:
- Recognize that \( \sqrt{-8\sqrt{7}} = \sqrt{8\sqrt{7}} \cdot i \).
- Simplifying \( \sqrt{8\sqrt{7}} \):
- \( \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \).
- Thus, \( \sqrt{8\sqrt{7}} = \sqrt{8} \cdot \sqrt{\sqrt{7}} = 2\sqrt{2 \sqrt{7}} = 2\sqrt{2} \cdot \sqrt{7}^{1/2} = 2\sqrt{2} \cdot 7^{1/4} = 2\sqrt{14} \).
- Therefore, it translates to \( -5\sqrt{7} - 2\sqrt{2}\) in real parts.
Now without the negativity sign, let's avoid complexities with \( i \) and compute value reverting back to the real domain as \( 5\sqrt{7} - \sqrt{-8\sqrt{7}} \).
However, if we focus on simplifying given you mentioned responses, let’s use given responses and try matching through expressions:
Essentially:
- \( 5\sqrt{7} + i(2\sqrt{14}) = 0 , \text{(for invalid cases)} \).
Switching back,
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- If a subtraction or modify to reveal the validity amongst simplifying leads us to gather.
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Thus, the responses you’ve given:
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Final Conclusion here leads towards:
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