To solve the equation [tex]\(\frac{1}{x^{-2}}=x^2\)[/tex], we can proceed step-by-step to discover the value of [tex]\(x\)[/tex].
1. Rewrite the equation:
[tex]\[
\frac{1}{x^{-2}} = x^2
\][/tex]
2. Simplify the fraction:
Recall that [tex]\(x^{-2}\)[/tex] can be rewritten using the property of exponents:
[tex]\[
x^{-2} = \frac{1}{x^2}
\][/tex]
Therefore, the equation becomes:
[tex]\[
\frac{1}{\frac{1}{x^2}} = x^2
\][/tex]
3. Simplify further:
[tex]\[
\frac{1}{\frac{1}{x^2}} = x^2 \implies x^2 = x^2
\][/tex]
4. Conclusion:
The simplified equation [tex]\(x^2 = x^2\)[/tex] is an identity that is true for all real numbers [tex]\(x\)[/tex] provided that [tex]\(x \neq 0\)[/tex] (since [tex]\(x = 0\)[/tex] would involve division by zero).
However, since we are solving for a variable [tex]\(x\)[/tex] within the provided equation and the result suggests all real numbers excluding zero, there are no specific solutions that satisfy the result as a unique value within typical algebraic constraints because any non-zero [tex]\(x\)[/tex] will make the original equation valid.
Therefore, the value [tex]\(z\)[/tex] does not have any unique solution as such in this context, resulting in the solution set being empty.
