Let's solve the given mathematical expression step-by-step:
We are given the expression [tex]\(\left(\frac{x^{-2}}{y^2}\right)^{\frac{1}{2}}\)[/tex] with [tex]\(x = -3\)[/tex] and [tex]\(y = 2\)[/tex].
1. Evaluate [tex]\(x^{-2}\)[/tex]:
Since [tex]\(x = -3\)[/tex],
[tex]\[
x^{-2} = (-3)^{-2}
\][/tex]
The exponent [tex]\(-2\)[/tex] means taking the reciprocal of [tex]\((-3)\)[/tex] squared:
[tex]\[
(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}
\][/tex]
2. Evaluate [tex]\(y^2\)[/tex]:
Since [tex]\(y = 2\)[/tex],
[tex]\[
y^2 = 2^2 = 4
\][/tex]
3. Form the fraction [tex]\(\frac{x^{-2}}{y^2}\)[/tex]:
Substitute the values we found into the fraction:
[tex]\[
\frac{x^{-2}}{y^2} = \frac{\frac{1}{9}}{4}
\][/tex]
4. Simplify the fraction:
Dividing [tex]\(\frac{1}{9}\)[/tex] by [tex]\(4\)[/tex] is the same as multiplying [tex]\(\frac{1}{9}\)[/tex] by the reciprocal of [tex]\(4\)[/tex]:
[tex]\[
\frac{\frac{1}{9}}{4} = \frac{1}{9} \times \frac{1}{4} = \frac{1}{36}
\][/tex]
5. Apply the exponent [tex]\(\frac{1}{2}\)[/tex]:
Now we need to take the square root of [tex]\(\frac{1}{36}\)[/tex] because raising to the power of [tex]\(\frac{1}{2}\)[/tex] is the same as taking the square root:
[tex]\[
\left(\frac{1}{36}\right)^{\frac{1}{2}} = \sqrt{\frac{1}{36}}
\][/tex]
6. Find the square root of [tex]\(\frac{1}{36}\)[/tex]:
[tex]\[
\sqrt{\frac{1}{36}} = \frac{1}{\sqrt{36}} = \frac{1}{6}
\][/tex]
So, the final answer is:
[tex]\[
\left(\frac{x^{-2}}{y^2}\right)^{\frac{1}{2}} = \frac{1}{6} \approx 0.16666666666666666
\][/tex]
