Step 1

Given:

\(\displaystyle{\frac{{{10}+\sqrt{{-{45}}}}}{{{4}}}}\)

Step 2

\(\displaystyle{\frac{{{10}+\sqrt{{-{45}}}}}{{{4}}}}={\frac{{{10}+{i}\sqrt{{{45}}}}}{{{4}}}}{\left(\sqrt{{-{1}}}={i}\right)}\)

\(\displaystyle={\frac{{{10}+{i}\sqrt{{{9}\times{5}}}}}{{{4}}}}\)

\(\displaystyle={\frac{{{10}+{i}{3}\sqrt{{{5}}}}}{{{4}}}}\)

Given:

\(\displaystyle{\frac{{{10}+\sqrt{{-{45}}}}}{{{4}}}}\)

Step 2

\(\displaystyle{\frac{{{10}+\sqrt{{-{45}}}}}{{{4}}}}={\frac{{{10}+{i}\sqrt{{{45}}}}}{{{4}}}}{\left(\sqrt{{-{1}}}={i}\right)}\)

\(\displaystyle={\frac{{{10}+{i}\sqrt{{{9}\times{5}}}}}{{{4}}}}\)

\(\displaystyle={\frac{{{10}+{i}{3}\sqrt{{{5}}}}}{{{4}}}}\)