Answer :

GinnyAnswer
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]

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