Section 6.4 Additional Practice Related to Imaginary and Complex Numbers
¶Exercises Exercises
1.
\(\sqrt{-81}\)
\(\sqrt{-81}=9i\)
2.
\(-\sqrt{-144}\)
\(-\sqrt{-144}=-12i\)
3.
\(\sqrt{-\frac{9}{25}}\)
\(\sqrt{-\frac{9}{25}}=\frac{3}{5}i\)
4.
\(-\sqrt{-90}\)
\(-\sqrt{-90}=-3\sqrt{10}i\)
5.
\(\sqrt{-325}\)
\(\sqrt{-325}=5\sqrt{13}i\)
6.
\(\sqrt{-\frac{27}{64}}\)
\(\sqrt{-\frac{27}{64}}=\frac{3\sqrt{3}}{8}i\)
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.
\((7i)(4i)\)
\((7i)(4i)=-28\)
8.
\((-2i)\left(\frac{3}{4}i\right)\)
\((-2i)\left(\frac{3}{4}i\right)=\frac{3}{2}\)
9.
\((-i)(-7i)\)
\((-i)(-7i)=-7\)
10.
\(\frac{3}{i}\)
\(\frac{3}{i}=-3i\)
11.
\(-\frac{5}{11i}\)
\(-\frac{5}{11i}=\frac{5}{11}i\)
12.
\((3i)(12i)\left(\frac{5}{6}i\right)\)
\((3i)(12i)\left(\frac{5}{6}i\right)=-30i\)
13.
\(i^{107}\)
\(i^{107}=-i\)
14.
\(i^{7981}\)
\(i^{7981}=i\)
15.
\(i^{-15}\)
\(i^{-15}=i\)
16.
\(i^{-14}\)
\(i^{-14}=-1\)
17.
\((5-4i)(-3+2i)\)
\((5-4i)(-3+2i)=-7+22i\)
18.
\((2-i)^2\)
\((2-i)^2=3-4i\)
19.
\((8-3i)(8+3i)\)
\((8-3i)(8+3i)=73\)
20.
\(\frac{24}{1-3i}\)
\(\frac{24}{1-3i}=\frac{12}{5}+\frac{36}{5}i\)
21.
\(-\frac{169i}{5+12i}\)
\(-\frac{169i}{5+12i}=-12-5i\)
22.
\(-\frac{1-2i}{1+2i}\)
\(-\frac{1-2i}{1+2i}=\frac{3}{5}+\frac{4}{5}i\)
23.
\(\frac{2+5i}{2-5i}\)
\(\frac{2+5i}{2-5i}=-\frac{21}{29}+\frac{20}{29}i\)
24.
\(\frac{-2-3i}{-2+3i}\)
\(\frac{-2-3i}{-2+3i}=-\frac{5}{13}+\frac{12}{13}i\)