Answer :
Sure, let's solve the equation [tex]\( x^8 - 25 = 0 \)[/tex] in a detailed, step-by-step manner.
1. Setting up the Equation:
Start with the given equation:
[tex]\[ x^8 - 25 = 0 \][/tex]
2. Isolating the Variable Term:
Add 25 to both sides of the equation to isolate the [tex]\( x^8 \)[/tex] term:
[tex]\[ x^8 = 25 \][/tex]
3. Solving for [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], we need to take the eighth root of both sides of the equation. Therefore, we obtain:
[tex]\[ x = \pm \sqrt[8]{25} \][/tex]
4. Expressing the Solutions:
The eighth root of 25 can be written in terms of a fourth root. Specifically, since [tex]\( 25 = 5^2 \)[/tex], we have:
[tex]\[ \sqrt[8]{25} = (5^2)^{1/8} = 5^{1/4} \][/tex]
Thus, the solutions for [tex]\( x \)[/tex] are:
[tex]\[ x = \pm 5^{1/4} \][/tex]
However, we have to consider the complex roots as well. The general expression for the nth roots of a number [tex]\( k \)[/tex] can be written as [tex]\( k^{1/n} \text{cis}\left(\frac{2 \pi k}{n}\right) \)[/tex], where [tex]\( \text{cis}(\theta) = \cos(\theta) + i \sin(\theta) \)[/tex]. For our problem where [tex]\( x = \sqrt[8]{25} \)[/tex], this means we have to consider all eight roots.
5. Finding All Roots:
Consider the complex roots using Euler’s formula [tex]\( e^{i \theta} \)[/tex] which gives us:
[tex]\[ x = \pm 5^{1/4} e^{i \frac{2k\pi}{8}} \quad \text{for} \; k = 0, 1, \ldots, 7 \][/tex]
Simplifying the [tex]\( e^{i \frac{2k\pi}{8}} \)[/tex] terms, you find the roots to be:
[tex]\[ x = 5^{1/4} e^{i \frac{2k\pi}{8}} \quad \text{or} \quad x = - 5^{1/4} e^{i \frac{2k\pi}{8}} \][/tex]
On computing these for appropriate integer values of [tex]\( k \)[/tex] from 0 to 7, we get the following distinct roots:
- [tex]\( x_1 = 5^{1/4} \)[/tex]
- [tex]\( x_2 = -5^{1/4} \)[/tex]
- [tex]\( x_3 = 5^{1/4} i \)[/tex]
- [tex]\( x_4 = -5^{1/4} i \)[/tex]
- [tex]\( x_5 = -\frac{\sqrt{2}5^{1/4}}{2} - i \frac{\sqrt{2}5^{1/4}}{2} \)[/tex]
- [tex]\( x_6 = -\frac{\sqrt{2}5^{1/4}}{2} + i \frac{\sqrt{2}5^{1/4}}{2} \)[/tex]
- [tex]\( x_7 = \frac{\sqrt{2}5^{1/4}}{2} - i \frac{\sqrt{2}5^{1/4}}{2} \)[/tex]
- [tex]\( x_8 = \frac{\sqrt{2}5^{1/4}}{2} + i \frac{\sqrt{2}5^{1/4}}{2} \)[/tex]
Thus, the solutions for the equation [tex]\( x^8 - 25 = 0 \)[/tex] are:
[tex]\[ \left\{-5^{1/4}, 5^{1/4}, -5^{1/4}i, 5^{1/4}i, -\frac{\sqrt{2}5^{1/4}}{2} - i \frac{\sqrt{2}5^{1/4}}{2}, -\frac{\sqrt{2}5^{1/4}}{2} + i \frac{\sqrt{2}5^{1/4}}{2}, \frac{\sqrt{2}5^{1/4}}{2} - i \frac{\sqrt{2}5^{1/4}}{2}, \frac{\sqrt{2}5^{1/4}}{2} + i \frac{\sqrt{2}5^{1/4}}{2} \right\} \][/tex]
1. Setting up the Equation:
Start with the given equation:
[tex]\[ x^8 - 25 = 0 \][/tex]
2. Isolating the Variable Term:
Add 25 to both sides of the equation to isolate the [tex]\( x^8 \)[/tex] term:
[tex]\[ x^8 = 25 \][/tex]
3. Solving for [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], we need to take the eighth root of both sides of the equation. Therefore, we obtain:
[tex]\[ x = \pm \sqrt[8]{25} \][/tex]
4. Expressing the Solutions:
The eighth root of 25 can be written in terms of a fourth root. Specifically, since [tex]\( 25 = 5^2 \)[/tex], we have:
[tex]\[ \sqrt[8]{25} = (5^2)^{1/8} = 5^{1/4} \][/tex]
Thus, the solutions for [tex]\( x \)[/tex] are:
[tex]\[ x = \pm 5^{1/4} \][/tex]
However, we have to consider the complex roots as well. The general expression for the nth roots of a number [tex]\( k \)[/tex] can be written as [tex]\( k^{1/n} \text{cis}\left(\frac{2 \pi k}{n}\right) \)[/tex], where [tex]\( \text{cis}(\theta) = \cos(\theta) + i \sin(\theta) \)[/tex]. For our problem where [tex]\( x = \sqrt[8]{25} \)[/tex], this means we have to consider all eight roots.
5. Finding All Roots:
Consider the complex roots using Euler’s formula [tex]\( e^{i \theta} \)[/tex] which gives us:
[tex]\[ x = \pm 5^{1/4} e^{i \frac{2k\pi}{8}} \quad \text{for} \; k = 0, 1, \ldots, 7 \][/tex]
Simplifying the [tex]\( e^{i \frac{2k\pi}{8}} \)[/tex] terms, you find the roots to be:
[tex]\[ x = 5^{1/4} e^{i \frac{2k\pi}{8}} \quad \text{or} \quad x = - 5^{1/4} e^{i \frac{2k\pi}{8}} \][/tex]
On computing these for appropriate integer values of [tex]\( k \)[/tex] from 0 to 7, we get the following distinct roots:
- [tex]\( x_1 = 5^{1/4} \)[/tex]
- [tex]\( x_2 = -5^{1/4} \)[/tex]
- [tex]\( x_3 = 5^{1/4} i \)[/tex]
- [tex]\( x_4 = -5^{1/4} i \)[/tex]
- [tex]\( x_5 = -\frac{\sqrt{2}5^{1/4}}{2} - i \frac{\sqrt{2}5^{1/4}}{2} \)[/tex]
- [tex]\( x_6 = -\frac{\sqrt{2}5^{1/4}}{2} + i \frac{\sqrt{2}5^{1/4}}{2} \)[/tex]
- [tex]\( x_7 = \frac{\sqrt{2}5^{1/4}}{2} - i \frac{\sqrt{2}5^{1/4}}{2} \)[/tex]
- [tex]\( x_8 = \frac{\sqrt{2}5^{1/4}}{2} + i \frac{\sqrt{2}5^{1/4}}{2} \)[/tex]
Thus, the solutions for the equation [tex]\( x^8 - 25 = 0 \)[/tex] are:
[tex]\[ \left\{-5^{1/4}, 5^{1/4}, -5^{1/4}i, 5^{1/4}i, -\frac{\sqrt{2}5^{1/4}}{2} - i \frac{\sqrt{2}5^{1/4}}{2}, -\frac{\sqrt{2}5^{1/4}}{2} + i \frac{\sqrt{2}5^{1/4}}{2}, \frac{\sqrt{2}5^{1/4}}{2} - i \frac{\sqrt{2}5^{1/4}}{2}, \frac{\sqrt{2}5^{1/4}}{2} + i \frac{\sqrt{2}5^{1/4}}{2} \right\} \][/tex]
