We appreciate your visit to Solve the equation tex 9x 7 1 0 tex. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To solve the equation [tex]\(9x^7 - 1 = 0\)[/tex], we want to find the values of [tex]\(x\)[/tex] that make the equation equal to zero.
Here are the steps to solve the equation:
1. Equation Setup:
Start with the given equation:
[tex]\[
9x^7 - 1 = 0
\][/tex]
2. Isolate the Power Term:
Add 1 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
9x^7 = 1
\][/tex]
3. Divide by 9:
Divide both sides by 9 to solve for [tex]\(x^7\)[/tex]:
[tex]\[
x^7 = \frac{1}{9}
\][/tex]
4. Take the Seventh Root:
To solve for [tex]\(x\)[/tex], take the seventh root of both sides. This will result in:
[tex]\[
x = \left(\frac{1}{9}\right)^{1/7}
\][/tex]
5. Complex Roots:
When solving for the seventh roots, we take into consideration that there are 7 distinct roots (one real and six complex). These correspond to the values:
- The real root: [tex]\( x = 3^{5/7}/3 \)[/tex]
- The complex roots, which are obtained using the formulas involving trigonometric expressions (cosine and sine terms), taking into account the angles distributed over a circle, which relate to the roots of unity. These complex roots are:
[tex]\[
x = -\frac{3^{5/7}}{3} \left(\cos\left(\frac{\pi}{7}\right) + i\sin\left(\frac{\pi}{7}\right)\right)
\][/tex]
[tex]\[
x = -\frac{3^{5/7}}{3} \left(\cos\left(\frac{\pi}{7}\right) - i\sin\left(\frac{\pi}{7}\right)\right)
\][/tex]
[tex]\[
x = \frac{3^{5/7}}{3} \left(\cos\left(\frac{2\pi}{7}\right) - i\sin\left(\frac{2\pi}{7}\right)\right)
\][/tex]
[tex]\[
x = \frac{3^{5/7}}{3} \left(\cos\left(\frac{2\pi}{7}\right) + i\sin\left(\frac{2\pi}{7}\right)\right)
\][/tex]
[tex]\[
x = -\frac{3^{5/7}}{3} \left(\cos\left(\frac{3\pi}{7}\right) + i\sin\left(\frac{3\pi}{7}\right)\right)
\][/tex]
[tex]\[
x = -\frac{3^{5/7}}{3} \left(\cos\left(\frac{3\pi}{7}\right) - i\sin\left(\frac{3\pi}{7}\right)\right)
\][/tex]
These represent all the solutions for the equation [tex]\(9x^7 - 1 = 0\)[/tex], including both the real and complex roots.
Here are the steps to solve the equation:
1. Equation Setup:
Start with the given equation:
[tex]\[
9x^7 - 1 = 0
\][/tex]
2. Isolate the Power Term:
Add 1 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
9x^7 = 1
\][/tex]
3. Divide by 9:
Divide both sides by 9 to solve for [tex]\(x^7\)[/tex]:
[tex]\[
x^7 = \frac{1}{9}
\][/tex]
4. Take the Seventh Root:
To solve for [tex]\(x\)[/tex], take the seventh root of both sides. This will result in:
[tex]\[
x = \left(\frac{1}{9}\right)^{1/7}
\][/tex]
5. Complex Roots:
When solving for the seventh roots, we take into consideration that there are 7 distinct roots (one real and six complex). These correspond to the values:
- The real root: [tex]\( x = 3^{5/7}/3 \)[/tex]
- The complex roots, which are obtained using the formulas involving trigonometric expressions (cosine and sine terms), taking into account the angles distributed over a circle, which relate to the roots of unity. These complex roots are:
[tex]\[
x = -\frac{3^{5/7}}{3} \left(\cos\left(\frac{\pi}{7}\right) + i\sin\left(\frac{\pi}{7}\right)\right)
\][/tex]
[tex]\[
x = -\frac{3^{5/7}}{3} \left(\cos\left(\frac{\pi}{7}\right) - i\sin\left(\frac{\pi}{7}\right)\right)
\][/tex]
[tex]\[
x = \frac{3^{5/7}}{3} \left(\cos\left(\frac{2\pi}{7}\right) - i\sin\left(\frac{2\pi}{7}\right)\right)
\][/tex]
[tex]\[
x = \frac{3^{5/7}}{3} \left(\cos\left(\frac{2\pi}{7}\right) + i\sin\left(\frac{2\pi}{7}\right)\right)
\][/tex]
[tex]\[
x = -\frac{3^{5/7}}{3} \left(\cos\left(\frac{3\pi}{7}\right) + i\sin\left(\frac{3\pi}{7}\right)\right)
\][/tex]
[tex]\[
x = -\frac{3^{5/7}}{3} \left(\cos\left(\frac{3\pi}{7}\right) - i\sin\left(\frac{3\pi}{7}\right)\right)
\][/tex]
These represent all the solutions for the equation [tex]\(9x^7 - 1 = 0\)[/tex], including both the real and complex roots.
Thanks for taking the time to read Solve the equation tex 9x 7 1 0 tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
