To solve the equation \(-x^4 - 2 = -10\), we can first simplify it.
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Start by adding 2 to both sides:
\(-x^4 - 2 + 2 = -10 + 2\)
This simplifies to:
\(-x^4 = -8\)
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Now, multiply both sides by -1:
\(x^4 = 8\)
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To find \(x\), take the fourth root of both sides:
\(x = \pm \sqrt[4]{8}\)
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We can simplify \(\sqrt[4]{8}\):
\(\sqrt[4]{8} = \sqrt[4]{2^3} = \sqrt{2^{3/2}} = \sqrt{2 \cdot \sqrt{2}} = \sqrt{2} \cdot 2^{1/4}\).
However, for practical purposes, we can also express this as:
\(\sqrt[4]{8} = \sqrt[4]{(2^2)(2)} = \sqrt{2} \cdot \sqrt[4]{2}\)
- Calculating \(\sqrt[4]{8}\) gives us approximately \(1.414^{1/2} \cdot 2^{1/4} \approx 1.68\).
Since none of these precisely correspond to the responses provided, we recognize either \(x = \sqrt[4]{8} \approx 1.68\) or its negative \(-\sqrt[4]{8}\) can represent potential answers.
Given the options provided:
- x = 32
- x = 48
- x = -2
- x = 3
None exactly correspond to \(\pm \sqrt[4]{8}\).
However, on double-checking the initial calculation:
In decimal values: \( \sqrt[4]{8} \approx \pm 1.68\) which does not match the options such as 32 or 48.
It seems there might be confusion with the values presented since 2 raised to the fourth power equals 16, leading me to suspect errors in the proposed values or an error in computational interpretation.
When asked to specify the numeric result based on all calculation methods, the solution suggests possible root forms reflecting given numeric representations matched from basic polynomial and quartile outputs leading to an enumeration context that necessitates looking beyond canonical equations provided earlier.
However, if other results such as -2 or 3 are special admissions to solution forms within output question interpretations, please denote them in recovery where possibly needed for clarifying output.
Thus, if adjusted consideration finds searching precedences possible, from roots or simplifying powers, please reiterate contexts desired, resolvable reflections in each excursion state aforementioned solutions converged.