Solve Two-Step Linear Equations Practice

Question

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Question
Solve for x
.
x4−2=−10
(1 point)
x=

Answer 1

To solve the equation \( x^4 - 2 = -10 \), follow these steps:

1. Add 2 to both sides of the equation: \[ x^4 - 2 + 2 = -10 + 2 \] \[ x^4 = -8 \]

2. At this point, we notice that \( x^4 \) cannot equal a negative number in the realm of real numbers (since any number raised to an even power is non-negative). Therefore, we can conclude that there are no real solutions.

However, if you are looking for complex solutions, we can continue:

3. Take the fourth root of both sides: \[ x = \sqrt[4]{-8} \]

4. Express \(-8\) as \(-8 = 8 \cdot e^{i\pi}\): \[ x = \sqrt[4]{8} \cdot \sqrt[4]{e^{i\pi}} = \sqrt[4]{8} \cdot \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) \] \[ = \sqrt[4]{8} \cdot \frac{\sqrt{2}}{2} + i \sqrt[4]{8} \cdot \frac{\sqrt{2}}{2} \]

The fourth roots of \(-8\) thus give four complex solutions. However, since you might be looking for a simple numerical answer instead, the conclusion is:

There are no real solutions for \( x \).