https://file.aiquickdraw.com/mathbot/file/1730134483298pe5jb5qw.jpg

The equation provided is:

\[
\left( \frac{1}{x^2} - \frac{3}{x^2} \right)^2 = x^2
\]

To solve it, let's go through each step.

### Step 1: Simplify the Expression Inside the Parentheses
Combine the terms inside the parentheses:
\[
\frac{1}{x^2} - \frac{3}{x^2} = \frac{1 - 3}{x^2} = \frac{-2}{x^2}
\]

So, the equation becomes:
\[
\left( \frac{-2}{x^2} \right)^2 = x^2
\]

### Step 2: Square Both Sides
Now, square \(\frac{-2}{x^2}\):
\[
\frac{(-2)^2}{(x^2)^2} = \frac{4}{x^4}
\]

Thus, the equation becomes:
\[
\frac{4}{x^4} = x^2
\]

### Step 3: Clear the Denominator
Multiply both sides by \(x^4\) to eliminate the fraction:
\[
4 = x^6
\]

### Step 4: Solve for \(x\)
To solve for \(x\), take the sixth root of both sides:
\[
x = \pm \sqrt[6]{4}
\]

Therefore, the solutions are:
\[
x = \pm \sqrt[6]{4}
\]

Would you like further explanation or have any questions?

### Related Questions

1. How can we solve equations involving higher powers of \(x\) more generally?
2. What does taking a root mean in complex cases, such as with even or odd roots?
3. What techniques can help simplify expressions involving fractions and powers?
4. How does squaring affect equations and what precautions should we take?
5. How does multiplying through by variables impact the solutions of an equation?

### Tip
Always check if simplifying expressions inside parentheses can reduce the complexity of the equation before proceeding with more steps.

Solve \( \left( \frac{1}{x^2} - \frac{3}{x^2} \right)^2 = x^2 \)

Learn to solve the equation \( \left( \frac{1}{x^2} - \frac{3}{x^2} \right)^2 = x^2 \) through step-by-step simplification. This example involves algebraic manipulation, exponent rules, and solving quadratic equations, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation