To solve the equation \( x^4 - 2 = -10 \), first, we need to isolate the term involving \( x \):
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Add 2 to both sides of the equation:
\[ x^4 - 2 + 2 = -10 + 2 \]
This simplifies to:
\[ x^4 = -8 \]
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Next, to solve for \( x \), we take the fourth root of both sides. However, note that since \( x^4 \) cannot be negative for real numbers, there will be complex solutions.
Rewrite -8 as: \[ x^4 = -8 = 8 \times (-1) \]
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To find \( x \), take the fourth root: \[ x = \sqrt[4]{8 \times (-1)} = \sqrt[4]{8} \cdot \sqrt[4]{-1} \]
We can simplify \( \sqrt[4]{8} \): \[ \sqrt[4]{8} = \sqrt[4]{2^3} = 2^{3/4} = 2^{0.75} \]
And \( \sqrt[4]{-1} = e^{i\pi/4} \) (considering the fourth roots of -1).
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The fourth roots of -1 are given by: \[ e^{i(\pi/4 + k\pi/2)} \quad \text{for} , k = 0, 1, 2, 3 \]
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Thus, the four solutions \( x \) are: \[ x = 2^{3/4} e^{i(\pi/4 + k\pi/2)} \quad \text{for} , k = 0, 1, 2, 3 \]
Calculating these, the values of \( k \) will give you:
- For \( k = 0 \): \( x_0 = 2^{3/4} e^{i\pi/4} \)
- For \( k = 1 \): \( x_1 = 2^{3/4} e^{i(3\pi/4)} \)
- For \( k = 2 \): \( x_2 = 2^{3/4} e^{i(5\pi/4)} \)
- For \( k = 3 \): \( x_3 = 2^{3/4} e^{i(7\pi/4)} \)
Therefore, the four complex solutions for \( x \) are: \[ x = 2^{3/4} \left( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i \right), \quad x = 2^{3/4} \left( -\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i \right), \quad x = 2^{3/4} \left( -\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} i \right), \quad x = 2^{3/4} \left( \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} i \right) \]
These are your final answers.
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