To solve the equation [tex]\(\left(\frac{x^{-1}}{y^{-4}}\right)^6 = \frac{x^2}{y^8}\)[/tex], we need to follow these steps:
### Step 1: Simplify the Left Side of the Equation
The left side of the equation is [tex]\(\left(\frac{x^{-1}}{y^{-4}}\right)^6\)[/tex].
Apply the power rule: [tex]\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)[/tex].
Thus, [tex]\(\left(\frac{x^{-1}}{y^{-4}}\right)^6 = \frac{(x^{-1})^6}{(y^{-4})^6}\)[/tex].
Simplify the exponents:
[tex]\[
(x^{-1})^6 = x^{-1 \cdot 6} = x^{-6}
\][/tex]
[tex]\[
(y^{-4})^6 = y^{-4 \cdot 6} = y^{-24}
\][/tex]
So the left side simplifies to:
[tex]\[
\frac{x^{-6}}{y^{-24}}
\][/tex]
### Step 2: Simplify the Right Side of the Equation
The right side of the equation is given as [tex]\(\frac{x^2}{y^8}\)[/tex].
### Step 3: Equate the Simplified Left and Right Sides
Now we have:
[tex]\[
\frac{x^{-6}}{y^{-24}} = \frac{x^2}{y^8}
\][/tex]
### Step 4: Simplify the Equation
To simplify further, use the property of negative exponents [tex]\(\frac{1}{a^{-n}} = a^n\)[/tex]:
[tex]\[
\frac{x^{-6}}{y^{-24}} = x^{-6} \cdot y^{24}
\][/tex]
[tex]\[
\frac{x^2}{y^8} = x^2 \cdot y^{-8}
\][/tex]
Thus, equating both sides, we get:
[tex]\[
x^{-6} \cdot y^{24} = x^2 \cdot y^{-8}
\][/tex]
### Step 5: Solving the Equation
For the equation to hold true, the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] must be equal on both sides.
Thus we can equate the exponents of [tex]\(x\)[/tex]:
[tex]\[
-6 = 2
\][/tex]
This is a contradiction because [tex]\(-6\)[/tex] does not equal [tex]\(2\)[/tex].
Similarly, equate the exponents of [tex]\(y\)[/tex]:
[tex]\[
24 = -8
\][/tex]
This is also a contradiction because [tex]\(24\)[/tex] does not equal [tex]\(-8\)[/tex].
### Conclusion
Given the above steps, the result is that the equation cannot be true with real numbers for the given exponents. Therefore, the equation [tex]\(\left(\frac{x^{-1}}{y^{-4}}\right)^6=\frac{x^2}{y^8}\)[/tex] requires further context or domain-specific details to solve accurately.
