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To solve the equation [tex]\(128 \cdot w^7 = 1\)[/tex], we proceed as follows:
1. Isolate [tex]\( w^7 \)[/tex]:
[tex]\[ 128 \cdot w^7 = 1 \][/tex]
Divide both sides by 128:
[tex]\[ w^7 = \frac{1}{128} \][/tex]
2. Simplify the right-hand side:
Recall that [tex]\(128 = 2^7\)[/tex], so we can write:
[tex]\[ w^7 = \frac{1}{2^7} = 2^{-7} \][/tex]
3. Take the seventh root of both sides:
We need to find the seventh roots of [tex]\(\frac{1}{128}\)[/tex] or [tex]\(2^{-7}\)[/tex]. The principal seventh root can be written as:
[tex]\[ w = (2^{-7})^{\frac{1}{7}} = 2^{-1} = \frac{1}{2} \][/tex]
4. Find all seventh roots:
Since [tex]\( w^7 = \frac{1}{128} \)[/tex] is a complex number, we use the fact that the roots of a complex number [tex]\(z\)[/tex] can be found using the form:
[tex]\[ w_k = r^{\frac{1}{n}} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right) \][/tex]
for [tex]\( k = 0, 1, 2, \ldots, 6 \)[/tex], where [tex]\(r\)[/tex] is the magnitude of the complex number and [tex]\(\theta\)[/tex] is the argument (angle).
For [tex]\(\frac{1}{128}\)[/tex], which has a magnitude of [tex]\(\left| \frac{1}{128} \right| = \frac{1}{128}\)[/tex], and an argument [tex]\(\theta = 0 \)[/tex] (since [tex]\(\frac{1}{128}\)[/tex] is on the real axis), the roots are:
[tex]\[ w_k = \left(\frac{1}{128}\right)^{\frac{1}{7}} \left( \cos\left(\frac{2k\pi}{7}\right) + i \sin\left(\frac{2k\pi}{7}\right) \right) \][/tex]
Simplifying the magnitude part gives us the principal root:
[tex]\[ (2^{-7})^{\frac{1}{7}} = 2^{-1} = \frac{1}{2} \][/tex]
Combining the magnitude and the angle for each [tex]\( k \)[/tex], we get:
[tex]\[ w_k = \frac{1}{2} \left( \cos\left(\frac{2k\pi}{7}\right) + i \sin\left(\frac{2k\pi}{7}\right) \right) \][/tex]
The seven complex roots can then be listed explicitly by replacing [tex]\( k \)[/tex] from 0 to 6 and evaluating the trigonometric functions:
- For [tex]\( k = 0 \)[/tex]:
[tex]\[ w_0 = \frac{1}{2} \left( \cos\left(0\right) + i \sin\left(0\right) \right) = \frac{1}{2} \left(1 + 0i\right) = \frac{1}{2} \][/tex]
- For [tex]\( k = 1 \)[/tex]:
[tex]\[ w_1 = \frac{1}{2} \left( \cos\left(\frac{2\pi}{7}\right) + i \sin\left(\frac{2\pi}{7}\right) \right) \][/tex]
- For [tex]\( k = 2 \)[/tex]:
[tex]\[ w_2 = \frac{1}{2} \left( \cos\left(\frac{4\pi}{7}\right) + i \sin\left(\frac{4\pi}{7}\right) \right) \][/tex]
- For [tex]\( k = 3 \)[/tex]:
[tex]\[ w_3 = \frac{1}{2} \left( \cos\left(\frac{6\pi}{7}\right) + i \sin\left(\frac{6\pi}{7}\right) \right) \][/tex]
- For [tex]\( k = 4 \)[/tex]:
[tex]\[ w_4 = \frac{1}{2} \left( \cos\left(\frac{8\pi}{7}\right) + i \sin\left(\frac{8\pi}{7}\right) \right) \][/tex]
- For [tex]\( k = 5 \)[/tex]:
[tex]\[ w_5 = \frac{1}{2} \left( \cos\left(\frac{10\pi}{7}\right) + i \sin\left(\frac{10\pi}{7}\right) \right) \][/tex]
- For [tex]\( k = 6 \)[/tex]:
[tex]\[ w_6 = \frac{1}{2} \left( \cos\left(\frac{12\pi}{7}\right) + i \sin\left(\frac{12\pi}{7}\right) \right) \][/tex]
Combining these, the seven roots of the equation [tex]\(128 w^7 = 1\)[/tex] in simplified form are:
[tex]\[ \boxed{\left\{ \frac{1}{2}, \frac{1}{2} \left( \cos\left(\frac{2\pi}{7}\right) + i \sin\left(\frac{2\pi}{7}\right) \right), \frac{1}{2} \left( \cos\left(\frac{4\pi}{7}\right) + i \sin\left(\frac{4\pi}{7}\right) \right), \frac{1}{2} \left( \cos\left(\frac{6\pi}{7}\right) + i \sin\left(\frac{6\pi}{7}\right) \right), \frac{1}{2} \left( \cos\left(\frac{8\pi}{7}\right) + i \sin\left(\frac{8\pi}{7}\right) \right), \frac{1}{2} \left( \cos\left(\frac{10\pi}{7}\right) + i \sin\left(\frac{10\pi}{7}\right) \right), \frac{1}{2} \left( \cos\left(\frac{12\pi}{7}\right) + i \sin\left(\frac{12\pi}{7}\right) \right) \right\} } \][/tex]
1. Isolate [tex]\( w^7 \)[/tex]:
[tex]\[ 128 \cdot w^7 = 1 \][/tex]
Divide both sides by 128:
[tex]\[ w^7 = \frac{1}{128} \][/tex]
2. Simplify the right-hand side:
Recall that [tex]\(128 = 2^7\)[/tex], so we can write:
[tex]\[ w^7 = \frac{1}{2^7} = 2^{-7} \][/tex]
3. Take the seventh root of both sides:
We need to find the seventh roots of [tex]\(\frac{1}{128}\)[/tex] or [tex]\(2^{-7}\)[/tex]. The principal seventh root can be written as:
[tex]\[ w = (2^{-7})^{\frac{1}{7}} = 2^{-1} = \frac{1}{2} \][/tex]
4. Find all seventh roots:
Since [tex]\( w^7 = \frac{1}{128} \)[/tex] is a complex number, we use the fact that the roots of a complex number [tex]\(z\)[/tex] can be found using the form:
[tex]\[ w_k = r^{\frac{1}{n}} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right) \][/tex]
for [tex]\( k = 0, 1, 2, \ldots, 6 \)[/tex], where [tex]\(r\)[/tex] is the magnitude of the complex number and [tex]\(\theta\)[/tex] is the argument (angle).
For [tex]\(\frac{1}{128}\)[/tex], which has a magnitude of [tex]\(\left| \frac{1}{128} \right| = \frac{1}{128}\)[/tex], and an argument [tex]\(\theta = 0 \)[/tex] (since [tex]\(\frac{1}{128}\)[/tex] is on the real axis), the roots are:
[tex]\[ w_k = \left(\frac{1}{128}\right)^{\frac{1}{7}} \left( \cos\left(\frac{2k\pi}{7}\right) + i \sin\left(\frac{2k\pi}{7}\right) \right) \][/tex]
Simplifying the magnitude part gives us the principal root:
[tex]\[ (2^{-7})^{\frac{1}{7}} = 2^{-1} = \frac{1}{2} \][/tex]
Combining the magnitude and the angle for each [tex]\( k \)[/tex], we get:
[tex]\[ w_k = \frac{1}{2} \left( \cos\left(\frac{2k\pi}{7}\right) + i \sin\left(\frac{2k\pi}{7}\right) \right) \][/tex]
The seven complex roots can then be listed explicitly by replacing [tex]\( k \)[/tex] from 0 to 6 and evaluating the trigonometric functions:
- For [tex]\( k = 0 \)[/tex]:
[tex]\[ w_0 = \frac{1}{2} \left( \cos\left(0\right) + i \sin\left(0\right) \right) = \frac{1}{2} \left(1 + 0i\right) = \frac{1}{2} \][/tex]
- For [tex]\( k = 1 \)[/tex]:
[tex]\[ w_1 = \frac{1}{2} \left( \cos\left(\frac{2\pi}{7}\right) + i \sin\left(\frac{2\pi}{7}\right) \right) \][/tex]
- For [tex]\( k = 2 \)[/tex]:
[tex]\[ w_2 = \frac{1}{2} \left( \cos\left(\frac{4\pi}{7}\right) + i \sin\left(\frac{4\pi}{7}\right) \right) \][/tex]
- For [tex]\( k = 3 \)[/tex]:
[tex]\[ w_3 = \frac{1}{2} \left( \cos\left(\frac{6\pi}{7}\right) + i \sin\left(\frac{6\pi}{7}\right) \right) \][/tex]
- For [tex]\( k = 4 \)[/tex]:
[tex]\[ w_4 = \frac{1}{2} \left( \cos\left(\frac{8\pi}{7}\right) + i \sin\left(\frac{8\pi}{7}\right) \right) \][/tex]
- For [tex]\( k = 5 \)[/tex]:
[tex]\[ w_5 = \frac{1}{2} \left( \cos\left(\frac{10\pi}{7}\right) + i \sin\left(\frac{10\pi}{7}\right) \right) \][/tex]
- For [tex]\( k = 6 \)[/tex]:
[tex]\[ w_6 = \frac{1}{2} \left( \cos\left(\frac{12\pi}{7}\right) + i \sin\left(\frac{12\pi}{7}\right) \right) \][/tex]
Combining these, the seven roots of the equation [tex]\(128 w^7 = 1\)[/tex] in simplified form are:
[tex]\[ \boxed{\left\{ \frac{1}{2}, \frac{1}{2} \left( \cos\left(\frac{2\pi}{7}\right) + i \sin\left(\frac{2\pi}{7}\right) \right), \frac{1}{2} \left( \cos\left(\frac{4\pi}{7}\right) + i \sin\left(\frac{4\pi}{7}\right) \right), \frac{1}{2} \left( \cos\left(\frac{6\pi}{7}\right) + i \sin\left(\frac{6\pi}{7}\right) \right), \frac{1}{2} \left( \cos\left(\frac{8\pi}{7}\right) + i \sin\left(\frac{8\pi}{7}\right) \right), \frac{1}{2} \left( \cos\left(\frac{10\pi}{7}\right) + i \sin\left(\frac{10\pi}{7}\right) \right), \frac{1}{2} \left( \cos\left(\frac{12\pi}{7}\right) + i \sin\left(\frac{12\pi}{7}\right) \right) \right\} } \][/tex]
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