Let's solve the problem step-by-step:
Given expression:
[tex]\[
\frac{3^{-2}}{\left(\frac{1}{3}\right)^{-3} \left(\frac{1}{9}\right)^{-2}} - \left(2^2 : \left(\frac{1}{2}\right)^{-2}\right)
\][/tex]
### Step 1: Simplify each part separately
#### Term [tex]\( 3^{-2} \)[/tex]
[tex]\[
3^{-2} = \frac{1}{3^2} = \frac{1}{9} \approx 0.1111
\][/tex]
#### Term [tex]\(\left(\frac{1}{3}\right)^{-3}\)[/tex]
[tex]\[
\left(\frac{1}{3}\right)^{-3} = 3^3 = 27.0
\][/tex]
#### Term [tex]\(\left(\frac{1}{9}\right)^{-2}\)[/tex]
[tex]\[
\left(\frac{1}{9}\right)^{-2} = 9^2 = 81.0
\][/tex]
### Step 2: Combine the terms inside the main division
To find the value of
[tex]\[
\frac{3^{-2}}{\left(\frac{1}{3}\right)^{-3} \left(\frac{1}{9}\right)^{-2}},
\][/tex]
First, multiply the denominators:
[tex]\[
\left(\frac{1}{3}\right)^{-3} \times \left(\frac{1}{9}\right)^{-2} = 27 \times 81 = 2187.0
\][/tex]
Then, divide the numerator by this product:
[tex]\[
\frac{\frac{1}{9}}{2187} = \frac{1}{9} \times \frac{1}{2187} \approx 5.0805 \times 10^{-5}
\][/tex]
### Step 3: Simplify the right side of expression
For
[tex]\[
2^2 : \left(\frac{1}{2}\right)^{-2},
\][/tex]
First, simplify the divisor:
[tex]\[
\left(\frac{1}{2}\right)^{-2} = 2^2 = 4
\][/tex]
Then,
[tex]\[
2^2 \div 4 = 4 \div 4 = 1
\][/tex]
### Step 4: Subtract the right side from the combined term
Substitute back into the expression:
[tex]\[
5.0805 \times 10^{-5} - 1
\][/tex]
Subtracting these gives:
[tex]\[
5.0805 \times 10^{-5} - 1 \approx -0.9999
\][/tex]
Thus, the final result is approximately:
[tex]\[
\boxed{-0.9999}
\][/tex]
