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To solve the equation [tex]\((x^4)^2 = x\)[/tex], let's go through a detailed, step-by-step process.
### Step 1: Simplify the Exponential Equation
First, simplify [tex]\((x^4)^2\)[/tex]:
[tex]\[ (x^4)^2 = x^{4 \cdot 2} = x^8 \][/tex]
So the equation [tex]\( (x^4)^2 = x \)[/tex] becomes:
[tex]\[ x^8 = x \][/tex]
### Step 2: Set the Equation to Zero
Rearrange the equation to set it to zero:
[tex]\[ x^8 - x = 0 \][/tex]
### Step 3: Factor Out the Common Term
Factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x(x^7 - 1) = 0 \][/tex]
This gives us two simpler equations to solve:
1. [tex]\(x = 0\)[/tex]
2. [tex]\(x^7 - 1 = 0\)[/tex]
### Step 4: Solve for [tex]\( x = 0 \)[/tex]
The first equation [tex]\(x = 0\)[/tex] gives us one solution:
[tex]\[ x = 0 \][/tex]
### Step 5: Solve [tex]\( x^7 - 1 = 0 \)[/tex]
The second equation [tex]\(x^7 - 1 = 0\)[/tex] can be solved by noting that it's a difference of powers:
[tex]\[ x^7 = 1 \][/tex]
### Step 6: Find the Roots of [tex]\( x^7 = 1 \)[/tex]
The equation [tex]\(x^7 = 1\)[/tex] represents the 7th roots of unity. These are complex numbers that satisfy the equation. The solutions are given by:
[tex]\[ x = e^{2k\pi i / 7} \quad \text{for} \quad k = 0, 1, 2, 3, 4, 5, 6 \][/tex]
### Step 7: Express the Roots of Unity in Trigonometric Form
Express these solutions using Euler's formula [tex]\( e^{i\theta} = \cos(\theta) + i\sin(\theta) \)[/tex]:
[tex]\[ x_k = \cos\left(\frac{2k\pi}{7}\right) + i\sin\left(\frac{2k\pi}{7}\right) \quad \text{for} \quad k = 0, 1, 2, 3, 4, 5, 6 \][/tex]
### Step 8: List All Solutions
Combining the solutions, the full set of solutions for the equation [tex]\( (x^4)^2 = x \)[/tex] is:
[tex]\[ x = 0, 1, e^{2\pi i / 7}, e^{4\pi i / 7}, e^{6\pi i / 7}, e^{8\pi i / 7}, e^{10\pi i / 7}, e^{12\pi i / 7} \][/tex]
Expressing these roots of unity in trigonometric forms:
[tex]\[ 0, 1, \cos\left(\frac{2\pi}{7}\right) + i\sin\left(\frac{2\pi}{7}\right), \cos\left(\frac{4\pi}{7}\right) + i\sin\left(\frac{4\pi}{7}\right), \cos\left(\frac{6\pi}{7}\right) + i\sin\left(\frac{6\pi}{7}\right), \][/tex]
[tex]\[ \cos\left(\frac{8\pi}{7}\right) + i\sin\left(\frac{8\pi}{7}\right), \cos\left(\frac{10\pi}{7}\right) + i\sin\left(\frac{10\pi}{7}\right), \cos\left(\frac{12\pi}{7}\right) + i\sin\left(\frac{12\pi}{7}\right) \][/tex]
Since cosine and sine are functions that have symmetry properties, some solutions simplify to their negative counterparts. Essentially, the solutions reflect the symmetry on the unit circle:
[tex]\[ 0, 1, -\cos\left(\frac{\pi}{7}\right) + i\sin\left(\frac{\pi}{7}\right), -\cos\left(\frac{\pi}{7}\right) - i\sin\left(\frac{\pi}{7}\right), \cos\left(\frac{2\pi}{7}\right) + i\sin\left(\frac{2\pi}{7}\right), \][/tex]
[tex]\[ \cos\left(\frac{2\pi}{7}\right) - i\sin\left(\frac{2\pi}{7}\right), -\cos\left(\frac{3\pi}{7}\right) + i\sin\left(\frac{3\pi}{7}\right), -\cos\left(\frac{3\pi}{7}\right) - i\sin\left(\frac{3\pi}{7}\right) \][/tex]
So, the solutions are:
[tex]\[ [0, 1, -\cos\left(\frac{\pi}{7}\right) - i\sin\left(\frac{\pi}{7}\right), -\cos\left(\frac{\pi}{7}\right) + i\sin\left(\frac{\pi}{7}\right), \cos\left(\frac{2\pi}{7}\right) - i\sin\left(\frac{2\pi}{7}\right), \cos\left(\frac{2\pi}{7}\right) + i\sin\left(\frac{2\pi}{7}\right), -\cos\left(\frac{3\pi}{7}\right) - i\sin\left(\frac{3\pi}{7}\right), -\cos\left(\frac{3\pi}{7}\right) + i\sin\left(\frac{3\pi}{7}\right)] \][/tex]
### Step 1: Simplify the Exponential Equation
First, simplify [tex]\((x^4)^2\)[/tex]:
[tex]\[ (x^4)^2 = x^{4 \cdot 2} = x^8 \][/tex]
So the equation [tex]\( (x^4)^2 = x \)[/tex] becomes:
[tex]\[ x^8 = x \][/tex]
### Step 2: Set the Equation to Zero
Rearrange the equation to set it to zero:
[tex]\[ x^8 - x = 0 \][/tex]
### Step 3: Factor Out the Common Term
Factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x(x^7 - 1) = 0 \][/tex]
This gives us two simpler equations to solve:
1. [tex]\(x = 0\)[/tex]
2. [tex]\(x^7 - 1 = 0\)[/tex]
### Step 4: Solve for [tex]\( x = 0 \)[/tex]
The first equation [tex]\(x = 0\)[/tex] gives us one solution:
[tex]\[ x = 0 \][/tex]
### Step 5: Solve [tex]\( x^7 - 1 = 0 \)[/tex]
The second equation [tex]\(x^7 - 1 = 0\)[/tex] can be solved by noting that it's a difference of powers:
[tex]\[ x^7 = 1 \][/tex]
### Step 6: Find the Roots of [tex]\( x^7 = 1 \)[/tex]
The equation [tex]\(x^7 = 1\)[/tex] represents the 7th roots of unity. These are complex numbers that satisfy the equation. The solutions are given by:
[tex]\[ x = e^{2k\pi i / 7} \quad \text{for} \quad k = 0, 1, 2, 3, 4, 5, 6 \][/tex]
### Step 7: Express the Roots of Unity in Trigonometric Form
Express these solutions using Euler's formula [tex]\( e^{i\theta} = \cos(\theta) + i\sin(\theta) \)[/tex]:
[tex]\[ x_k = \cos\left(\frac{2k\pi}{7}\right) + i\sin\left(\frac{2k\pi}{7}\right) \quad \text{for} \quad k = 0, 1, 2, 3, 4, 5, 6 \][/tex]
### Step 8: List All Solutions
Combining the solutions, the full set of solutions for the equation [tex]\( (x^4)^2 = x \)[/tex] is:
[tex]\[ x = 0, 1, e^{2\pi i / 7}, e^{4\pi i / 7}, e^{6\pi i / 7}, e^{8\pi i / 7}, e^{10\pi i / 7}, e^{12\pi i / 7} \][/tex]
Expressing these roots of unity in trigonometric forms:
[tex]\[ 0, 1, \cos\left(\frac{2\pi}{7}\right) + i\sin\left(\frac{2\pi}{7}\right), \cos\left(\frac{4\pi}{7}\right) + i\sin\left(\frac{4\pi}{7}\right), \cos\left(\frac{6\pi}{7}\right) + i\sin\left(\frac{6\pi}{7}\right), \][/tex]
[tex]\[ \cos\left(\frac{8\pi}{7}\right) + i\sin\left(\frac{8\pi}{7}\right), \cos\left(\frac{10\pi}{7}\right) + i\sin\left(\frac{10\pi}{7}\right), \cos\left(\frac{12\pi}{7}\right) + i\sin\left(\frac{12\pi}{7}\right) \][/tex]
Since cosine and sine are functions that have symmetry properties, some solutions simplify to their negative counterparts. Essentially, the solutions reflect the symmetry on the unit circle:
[tex]\[ 0, 1, -\cos\left(\frac{\pi}{7}\right) + i\sin\left(\frac{\pi}{7}\right), -\cos\left(\frac{\pi}{7}\right) - i\sin\left(\frac{\pi}{7}\right), \cos\left(\frac{2\pi}{7}\right) + i\sin\left(\frac{2\pi}{7}\right), \][/tex]
[tex]\[ \cos\left(\frac{2\pi}{7}\right) - i\sin\left(\frac{2\pi}{7}\right), -\cos\left(\frac{3\pi}{7}\right) + i\sin\left(\frac{3\pi}{7}\right), -\cos\left(\frac{3\pi}{7}\right) - i\sin\left(\frac{3\pi}{7}\right) \][/tex]
So, the solutions are:
[tex]\[ [0, 1, -\cos\left(\frac{\pi}{7}\right) - i\sin\left(\frac{\pi}{7}\right), -\cos\left(\frac{\pi}{7}\right) + i\sin\left(\frac{\pi}{7}\right), \cos\left(\frac{2\pi}{7}\right) - i\sin\left(\frac{2\pi}{7}\right), \cos\left(\frac{2\pi}{7}\right) + i\sin\left(\frac{2\pi}{7}\right), -\cos\left(\frac{3\pi}{7}\right) - i\sin\left(\frac{3\pi}{7}\right), -\cos\left(\frac{3\pi}{7}\right) + i\sin\left(\frac{3\pi}{7}\right)] \][/tex]
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