Let's go through the calculations step-by-step.
1. For the first expression, [tex]\(32^{-\frac{3}{5}}\)[/tex]:
[tex]\[ 32^{-\frac{3}{5}} = \left( \frac{1}{32} \right)^{\frac{3}{5}} \][/tex]
First, observe that 32 can be written as [tex]\(2^5\)[/tex]:
[tex]\[ 32 = 2^5 \][/tex]
Thus,
[tex]\[ 32^{-\frac{3}{5}} = (2^5)^{-\frac{3}{5}} \][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ (2^5)^{-\frac{3}{5}} = 2^{5 \cdot -\frac{3}{5}} = 2^{-3} \][/tex]
Now, calculate [tex]\(2^{-3}\)[/tex]:
[tex]\[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125 \][/tex]
So,
[tex]\[ 32^{-\frac{3}{5}} = 0.125 \][/tex]
2. For the second expression, [tex]\(\left(\frac{1}{9}\right)^{\frac{3}{2}}\)[/tex]:
Rewrite [tex]\(1/9\)[/tex] as [tex]\(9^{-1}\)[/tex]:
[tex]\[ \left(\frac{1}{9}\right)^{\frac{3}{2}} = \left(9^{-1}\right)^{\frac{3}{2}} \][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ \left(9^{-1}\right)^{\frac{3}{2}} = 9^{-\frac{3}{2}} \][/tex]
Now express 9 as [tex]\(3^2\)[/tex]:
[tex]\[ 9 = 3^2 \][/tex]
Thus,
[tex]\[ 9^{-\frac{3}{2}} = (3^2)^{-\frac{3}{2}} \][/tex]
Applying the property of exponents again:
[tex]\[ (3^2)^{-\frac{3}{2}} = 3^{2 \cdot -\frac{3}{2}} = 3^{-3} \][/tex]
Now, calculate [tex]\(3^{-3}\)[/tex]:
[tex]\[ 3^{-3} = \frac{1}{3^3} = \frac{1}{27} \approx 0.037037037037037035 \][/tex]
So,
[tex]\[ \left(\frac{1}{9}\right)^{\frac{3}{2}} \approx 0.037037037037037035 \][/tex]
In summary, we have:
[tex]\[
32^{-\frac{3}{5}} = 0.125
\][/tex]
[tex]\[
\left(\frac{1}{9}\right)^{\frac{3}{2}} \approx 0.037037037037037035
\][/tex]
