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Section 5.4 Additional Practice Related to Imaginary and Complex Numbers

Exercises Exercises

1.

\(\sqrt{-81}\)

Solution

\(\sqrt{-81}=9i\)

2.

\(-\sqrt{-144}\)

Solution

\(-\sqrt{-144}=-12i\)

3.

\(\sqrt{-\frac{9}{25}}\)

Solution

\(\sqrt{-\frac{9}{25}}=\frac{3}{5}i\)

4.

\(-\sqrt{-90}\)

Solution

\(-\sqrt{-90}=-3\sqrt{10}i\)

5.

\(\sqrt{-325}\)

Solution

\(\sqrt{-325}=5\sqrt{13}i\)

6.

\(\sqrt{-\frac{27}{64}}\)

Solution

\(\sqrt{-\frac{27}{64}}=\frac{3\sqrt{3}}{8}i\)

Simplify each of the following expressions. Please note that your final expression should be of the form \(a\text{,}\) \(bi\text{,}\) or \(a+bi\) where \(a\) and/or \(b\) are real number(s).

7.

\((7i)(4i)\)

Solution

\((7i)(4i)=-28\)

8.

\((-2i)\left(\frac{3}{4}i\right)\)

Solution

\((-2i)\left(\frac{3}{4}i\right)=\frac{3}{2}\)

9.

\((-i)(-7i)\)

Solution

\((-i)(-7i)=-7\)

10.

\(\frac{3}{i}\)

Solution

\(\frac{3}{i}=-3i\)

11.

\(-\frac{5}{11i}\)

Solution

\(-\frac{5}{11i}=\frac{5}{11}i\)

12.

\((3i)(12i)\left(\frac{5}{6}i\right)\)

Solution

\((3i)(12i)\left(\frac{5}{6}i\right)=-30i\)

14.

\(i^{7981}\)

Solution

\(i^{7981}=i\)

17.

\((5-4i)(-3+2i)\)

Solution

\((5-4i)(-3+2i)=-7+22i\)

18.

\((2-i)^2\)

Solution

\((2-i)^2=3-4i\)

19.

\((8-3i)(8+3i)\)

Solution

\((8-3i)(8+3i)=73\)

20.

\(\frac{24}{1-3i}\)

Solution

\(\frac{24}{1-3i}=\frac{12}{5}+\frac{36}{5}i\)

21.

\(-\frac{169i}{5+12i}\)

Solution

\(-\frac{169i}{5+12i}=-12-5i\)

22.

\(-\frac{1-2i}{1+2i}\)

Solution

\(-\frac{1-2i}{1+2i}=\frac{3}{5}+\frac{4}{5}i\)

23.

\(\frac{2+5i}{2-5i}\)

Solution

\(\frac{2+5i}{2-5i}=-\frac{21}{29}+\frac{20}{29}i\)

24.

\(\frac{-2-3i}{-2+3i}\)

Solution

\(\frac{-2-3i}{-2+3i}=-\frac{5}{13}+\frac{12}{13}i\)