Let’s solve these problems step-by-step.
For part (i):
Given expression:
[tex]\[
\sqrt{\frac{5}{6}} + \frac{-6}{5} + \frac{4}{10} + \frac{-7}{6}
\][/tex]
First, simplify each term:
[tex]\[
\sqrt{\frac{5}{6}}
\][/tex]
[tex]\(\sqrt{\frac{5}{6}}\)[/tex] is already simplified.
[tex]\[
\frac{-6}{5}
\][/tex]
[tex]\(\frac{-6}{5}\)[/tex] is already in its simplest form.
[tex]\[
\frac{4}{10}
\][/tex]
This simplifies to:
[tex]\[
\frac{4}{10} = \frac{2}{5}
\][/tex]
[tex]\[
\frac{-7}{6}
\][/tex]
[tex]\(\frac{-7}{6}\)[/tex] is already simplified.
Now, we sum these terms:
[tex]\[
\sqrt{\frac{5}{6}} + \frac{-6}{5} + \frac{2}{5} + \frac{-7}{6}
\][/tex]
Combining the like terms involves considering common denominators and simplifying:
The terms combine to yield [tex]\( -1.0537957374913898 \)[/tex].
For part (ii):
Given expression:
[tex]\[
\frac{3}{8} + \frac{5}{12} + \frac{5}{7} + \frac{7}{12}
\][/tex]
First, simplify each term:
[tex]\[
\frac{3}{8}
\][/tex]
[tex]\(\frac{3}{8}\)[/tex] is already simplified.
[tex]\[
\frac{5}{12}
\][/tex]
[tex]\(\frac{5}{12}\)[/tex] is already simplified.
[tex]\[
\frac{5}{7}
\][/tex]
[tex]\(\frac{5}{7}\)[/tex] is already simplified.
[tex]\[
\frac{7}{12}
\][/tex]
[tex]\(\frac{7}{12}\)[/tex] is already simplified.
Now, we sum these terms:
[tex]\[
\frac{3}{8}, \frac{5}{12}, \frac{5}{7}, \frac{7}{12}
\][/tex]
Combining the like terms, with appropriate common denominators and simplifications, yields [tex]\( 2.0892857142857144 \)[/tex].
So, the final sums are:
(i) [tex]\(-1.0537957374913898\)[/tex]
(ii) [tex]\(2.0892857142857144\)[/tex]
