Let's simplify each part step by step.
Part (i): Simplify [tex]\(\left[7^{-1} + \left(\frac{3}{2}\right)^{-1}\right]^{-1}\)[/tex]
1. First, simplify the individual terms inside the bracket:
- [tex]\(7^{-1} = \frac{1}{7}\)[/tex]
- [tex]\(\left(\frac{3}{2}\right)^{-1}\)[/tex] means taking the reciprocal of [tex]\(\frac{3}{2}\)[/tex], so [tex]\(\left(\frac{3}{2}\right)^{-1} = \frac{2}{3}\)[/tex]
2. Add these two fractions:
[tex]\[
\frac{1}{7} + \frac{2}{3} = \frac{3 \cdot 1 + 2 \cdot 7}{3 \cdot 7} = \frac{3 + 14}{21} = \frac{17}{21}
\][/tex]
3. Now, take the reciprocal of the result:
[tex]\[
\left(\frac{17}{21}\right)^{-1} = \frac{21}{17}
\][/tex]
Thus, the simplified form of [tex]\(\left[7^{-1} + \left(\frac{3}{2}\right)^{-1}\right]^{-1}\)[/tex] is [tex]\(\frac{21}{17}\)[/tex].
Numerically, the simplified value of [tex]\(\frac{21}{17}\)[/tex] is approximately [tex]\(1.2352941176470589\)[/tex].
Part (ii): Simplify [tex]\(\left(5^{-1} - 4^{-1}\right)^{-1} + \left(2^{-1} - 3^{-1}\right)^{-1}\)[/tex]
Let's break this into two separate components and simplify them individually.
First Component: [tex]\(\left(5^{-1} - 4^{-1}\right)^{-1}\)[/tex]
1. Simplify the individual terms:
- [tex]\(5^{-1} = \frac{1}{5}\)[/tex]
- [tex]\(4^{-1} = \frac{1}{4}\)[/tex]
2. Subtract the fractions:
[tex]\[
\frac{1}{5} - \frac{1}{4} = \frac{4 \cdot 1 - 5 \cdot 1}{5 \cdot 4} = \frac{4 - 5}{20} = \frac{-1}{20}
\][/tex]
3. Take the reciprocal of the result:
[tex]\[
\left(\frac{-1}{20}\right)^{-1} = -20
\][/tex]
Thus, the simplified form of [tex]\(\left(5^{-1} - 4^{-1}\right)^{-1}\)[/tex] is [tex]\(-20\)[/tex].
Second Component: [tex]\(\left(2^{-1} - 3^{-1}\right)^{-1}\)[/tex]
1. Simplify the individual terms:
- [tex]\(2^{-1} = \frac{1}{2}\)[/tex]
- [tex]\(3^{-1} = \frac{1}{3}\)[/tex]
2. Subtract the fractions:
[tex]\[
\frac{1}{2} - \frac{1}{3} = \frac{3 \cdot 1 - 2 \cdot 1}{3 \cdot 2} = \frac{3 - 2}{6} = \frac{1}{6}
\][/tex]
3. Take the reciprocal of the result:
[tex]\[
\left(\frac{1}{6}\right)^{-1} = 6
\][/tex]
Thus, the simplified form of [tex]\(\left(2^{-1} - 3^{-1}\right)^{-1}\)[/tex] is [tex]\(6\)[/tex].
Now, add the two components together:
[tex]\[
-20 + 6 = -14
\][/tex]
Thus, the simplified form of [tex]\(\left(5^{-1} - 4^{-1}\right)^{-1} + \left(2^{-1} - 3^{-1}\right)^{-1}\)[/tex] is [tex]\(-14\)[/tex].
### Summary of Simplified Results:
(i) [tex]\(\left[7^{-1} + \left(\frac{3}{2}\right)^{-1}\right]^{-1} \approx 1.2352941176470589\)[/tex]
(ii) [tex]\(\left(5^{-1} - 4^{-1}\right)^{-1} + \left(2^{-1} - 3^{-1}\right)^{-1} = -14\)[/tex]
So, the final result is:
[tex]\[
\left(1.2352941176470589, -20, 6, -14\right)
\][/tex]
