Let's solve the expression step-by-step:
[tex]\[ E = \sqrt[3]{\left[\left(\frac{1}{3}\right)^2 + \left(\frac{3}{2}\right)^{-2}\right]^{-1} - \left(1 \frac{1}{4}\right)^{-1}} \][/tex]
1. Calculate [tex]\(\left(\frac{1}{3}\right)^2\)[/tex]:
[tex]\[
\left(\frac{1}{3}\right)^2 = \frac{1}{9} \approx 0.1111
\][/tex]
2. Calculate [tex]\(\left(\frac{3}{2}\right)^{-2}\)[/tex]:
[tex]\[
\left(\frac{3}{2}\right)^{-2} = \left(\frac{2}{3}\right)^{2} = \frac{4}{9} \approx 0.4444
\][/tex]
3. Add the results of step 1 and step 2:
[tex]\[
\left(\frac{1}{3}\right)^2 + \left(\frac{3}{2}\right)^{-2} = \frac{1}{9} + \frac{4}{9} = \frac{5}{9} \approx 0.5556
\][/tex]
4. Take the inverse of this inner sum:
[tex]\[
\left(\frac{5}{9}\right)^{-1} = \frac{9}{5} \approx 1.8
\][/tex]
5. Convert the mixed fraction [tex]\(\left(1 \frac{1}{4}\right)\)[/tex] to an improper fraction:
[tex]\[
1 \frac{1}{4} = 1 + \frac{1}{4} = \frac{5}{4} = 1.25
\][/tex]
6. Invert this improper fraction:
[tex]\[
\left(\frac{5}{4}\right)^{-1} = \frac{4}{5} = 0.8
\][/tex]
7. Subtract the result of step 6 from the result of step 4:
[tex]\[
1.8 - 0.8 = 1.0
\][/tex]
8. Finally, take the cube root of the result from step 7:
[tex]\[
\sqrt[3]{1.0} = 1.0
\][/tex]
Thus, the value of [tex]\(E\)[/tex] is:
[tex]\[
E = 1.0
\][/tex]
