https://file.aiquickdraw.com/mathbot/file/173027920059664uo7n67.jpg

The equation shown is:

\[
\frac{1}{x^2} - 4x^4 = 0
\]

To solve this equation for \( x \), let's go through it step-by-step.

1. **Rewrite the Equation**: 
   Move the second term to the other side:
   \[
   \frac{1}{x^2} = 4x^4
   \]

2. **Clear the Denominator**:
   Multiply both sides by \( x^2 \) (assuming \( x \neq 0 \)) to eliminate the fraction:
   \[
   1 = 4x^6
   \]

3. **Isolate \( x^6 \)**:
   Divide both sides by 4:
   \[
   x^6 = \frac{1}{4}
   \]

4. **Solve for \( x \)**:
   Take the sixth root of both sides. Since we are dealing with a real number solution, we have:
   \[
   x = \pm \sqrt[6]{\frac{1}{4}}
   \]

5. **Simplify**:
   \[
   x = \pm \frac{1}{\sqrt[6]{4}} = \pm \frac{1}{2^{1/3}}
   \]

Thus, the solutions are:
\[
x = \pm \frac{1}{2^{1/3}}
\]

Would you like further details on any of these steps?

---

Here are some related questions you might explore:

1. How can we generalize this approach to equations involving higher powers?
2. What if the equation had a different constant instead of 4?
3. Can we solve similar equations by taking logarithms?
4. How would complex solutions differ in this problem?
5. What is the significance of rational exponents when solving polynomial equations?

**Tip:** When dealing with equations with fractions, clearing the denominator early on can simplify the problem.

Solve the equation \( \frac{1}{x^2} - 4x^4 = 0 \).

Learn to solve the rational equation \( \frac{1}{x^2} - 4x^4 = 0 \) with a step-by-step guide. This solution covers algebraic methods for isolating variables, clearing denominators, and solving equations with fractional exponents. Suitable for students in Grades 10-12.

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