Radicals
Simplify each radical.
1) \(\sqrt{50}\)
2) \(\sqrt{60}\)
3) \(\sqrt{40}\cdot\sqrt{2}\)
4) \(2\sqrt{15} \cdot5\sqrt{3}\)
5) \(7\sqrt{5}\cdot2\sqrt{10}\)
6) \(\sqrt{\dfrac{16}{25}}\)
7) \(\sqrt{\dfrac{6}{27}}\)
8) \(\dfrac{10}{\sqrt{3}}\)
9) \(\dfrac{32}{\sqrt{2}}\)
10) \(\sqrt{\dfrac{2}{3}}\)
11) \(\sqrt{\dfrac{16}{5}}\)
12) Which of the following are equivalent to \(10\sqrt{3}\)?
\(\:\:\dfrac{30}{\sqrt{3}}\:\:\) \(\:\:10\sqrt{9}\:\:\) \(\:\:\sqrt{300}\:\:\) \(\:\:3\sqrt{100}\:\:\) \(\:\:\sqrt{30}\cdot\sqrt{10}\:\:\) \(\:\:\sqrt{\dfrac{30}{9}}\:\:\)
13) Which of the following are equivalent to \(6\sqrt{2}\)?
\(\:\:\sqrt{\dfrac{6}{2}}\:\:\) \(\:\:\sqrt{3}\:\:\) \(\:\:\sqrt{72}\:\:\) \(\:\:\dfrac{12}{\sqrt{2}}\:\:\) \(\:\:\sqrt{\dfrac{36}{2}}\:\:\) \(\:\:\sqrt{8}\cdot\sqrt{9}\:\:\)
Simplify the following:
14) \(\dfrac{7}{(5+\sqrt{3})}\)
15) \(\dfrac{2}{(3-\sqrt{5})}\)
16) \(\dfrac{4}{(8-\sqrt{6})}\)
17) \(\sqrt{3}+6\sqrt{3}\)
18) \(\sqrt{45}-2\sqrt{5}\)
19) \(5\sqrt{32}+4\sqrt{8}\)
Complex Numbers
Simplify using Complex numbers.
20) \(\sqrt{-45}\)
21) \(\large\sqrt{-\frac{100}{81}}\)
22) \(\large\sqrt{-\frac{9}{7}}\)
23) \(2\sqrt{-40}\)
24) \(5i\sqrt{-54}\)
25) \((2+5i)+(4-9i)\)
26) \(6i(4+3i)\)
27) \(2(3+12i)-5(4+3i)\)
28) \((10-3i)(8+i)\)
29) \((5+2i)(5-2i)\)
30) \(\dfrac{5}{(3+i)}\)
31) \(\dfrac{(4-3i)}{(2+6i)}\)
Solution Bank
Simplify each radical.
1) \(\sqrt{50}\)
2) \(\sqrt{60}\)
3) \(\sqrt{40}\cdot\sqrt{2}\)
4) \(2\sqrt{15} \cdot5\sqrt{3}\)
5) \(7\sqrt{5}\cdot2\sqrt{10}\)
6) \(\sqrt{\dfrac{16}{25}}\)
7) \(\sqrt{\dfrac{6}{27}}\)
8) \(\dfrac{10}{\sqrt{3}}\)
9) \(\dfrac{32}{\sqrt{2}}\)
10) \(\sqrt{\dfrac{2}{3}}\)
11) \(\sqrt{\dfrac{16}{5}}\)
12) Which of the following are equivalent to \(10\sqrt{3}\)?
\(\:\:\dfrac{30}{\sqrt{3}}\:\:\) \(\:\:10\sqrt{9}\:\:\) \(\:\:\sqrt{300}\:\:\) \(\:\:3\sqrt{100}\:\:\) \(\:\:\sqrt{30}\cdot\sqrt{10}\:\:\) \(\:\:\sqrt{\dfrac{30}{9}}\:\:\)
13) Which of the following are equivalent to \(6\sqrt{2}\)?
\(\:\:\sqrt{\dfrac{6}{2}}\:\:\) \(\:\:\sqrt{3}\:\:\) \(\:\:\sqrt{72}\:\:\) \(\:\:\dfrac{12}{\sqrt{2}}\:\:\) \(\:\:\sqrt{\dfrac{36}{2}}\:\:\) \(\:\:\sqrt{8}\cdot\sqrt{9}\:\:\)
Simplify the following:
14) \(\dfrac{7}{(5+\sqrt{3})}\)
15) \(\dfrac{2}{(3-\sqrt{5})}\)
16) \(\dfrac{4}{(8-\sqrt{6})}\)
17) \(\sqrt{3}+6\sqrt{3}\)
18) \(\sqrt{45}-2\sqrt{5}\)
19) \(5\sqrt{32}+4\sqrt{8}\)
Complex Numbers
Simplify using Complex numbers.
20) \(\sqrt{-45}\)
21) \(\large\sqrt{-\frac{100}{81}}\)
22) \(\large\sqrt{-\frac{9}{7}}\)
23) \(2\sqrt{-40}\)
24) \(5i\sqrt{-54}\)
25) \((2+5i)+(4-9i)\)
26) \(6i(4+3i)\)
27) \(2(3+12i)-5(4+3i)\)
28) \((10-3i)(8+i)\)
29) \((5+2i)(5-2i)\)
30) \(\dfrac{5}{(3+i)}\)
31) \(\dfrac{(4-3i)}{(2+6i)}\)
Solution Bank