Radicals and Complex Numbers Review
Topic
Review
m n
 a mn
babg  a b
n
m
m
a m  a n  a mn
ca h
FG a IJ  FG a IJ
H bK H b K
m
Rules of
Exponents
n n
Test 2
Practice problems
1.
am
 a m n
n
a
a
n
1
 n
a
2.
(𝑥 3 𝑦𝑧 4 )2
𝑥 6 𝑦 −2 𝑧 5 𝑦 4 𝑧 3
45𝑎−3 𝑏6
(3𝑎4 )2
Name: ____________________________
FG ab IJ
H2K
2
=
3.
=
4.
2
=
a 2 (r 3 s) 2
=
a 5r 3 s 3
a0  1
3
7. 5𝑎 √−16𝑎10 𝑏5 =
- Write in terms of perfect
squares, cubes,etc.
- Root tells you how many
needed in a group to bring
outside of the radical.
- For variables, divide the
exponent by the root.
- Only use i on square
roots! Cube roots can be
negative.
- Factor trinomials to find
pairs!!
- Must have the same root
and radicand.
- Simplify all radicals.
- Combine like terms with
the same radicand (and
sometimes same variables
outside and inside the
radical).
5. - 3  27 =
10. 2√50 + 4√500 − 6√125 =
11. 2𝑥 2 √20𝑥 + 𝑥√125𝑥 3 − √45𝑥 5 =
Multiplying
Radicals
- Multiply coefficients
together.
- Multiply radicands
together.
- Simplify your answer.
12. (3√12)(3√20) =
13. 3𝑎𝑏 2 √18𝑎3 𝑏 ∙ 5𝑏√10𝑎 =
Multiplying
Radicals cont.
- Distribute or FOIL.
- Multiply coefficients
together and multiply
radicands together.
- Simplify your answer.
14. 43 18 53 4  3 12 =
16. (√2 − 3√6) =
15. 2√3𝑥(2𝑥√15𝑥 + 3√60𝑥 3 )
17. 7  5 3  2 5 =
Simplifying
Radicals
Adding and
Subtracting
Radicals
Dividing
Radicals
-Rationalize the
denominator.
-Simplify your answer.
6.
9.
d
18.
√
8𝑥 3
3𝑦
i
√𝑦 2
√𝑥𝑦𝑧 6
3
( x  4)15 =
2
d
id
3
=
20.
3
19. 3
x 2  6x  9 
8.
72a 8b15
=
i
3
√16𝑥 3 𝑦5
=
−6− √10
Dividing
Radicals cont.
- Multiply by the conjugate
of the denominator.
Complex
Numbers
√−1 = 𝑖
𝑖 2 = −1
𝑖 3 = −𝑖
𝑖4 = 1
- Final answer should not
have an i raised to a power.
23. (𝑖 5 )3 =
- Two parts: 1 part real and
1 part imaginary.
- Takes the form 𝑎 + 𝑏𝑖 .
- Final answer should not
have an i raised to a power.
- Denominator cannot have
an i.
26. (5 − 𝑖) − (3 − 2𝑖) =
1) Isolate the radical.
2) Square/cube both sides.
3) Solve for the variable.
4) Check your answer by
plugging back into orginal.
May have extraneous
solutions.
31. 18 − √7ℎ = 12
Solving
Equations
with complex
solutions
-Isolate the x2
-Take the √ , remember
the ±
34. 2x2 + 40 = 0
35. 8x2 + 5 = 1
Finding the
variables for
the real and
complex parts
of an equation
Graphing
-Set up a complex
equation.
-Set up a real number
equation.
-Solve each.
𝑦 = 𝑎√𝑥 − ℎ + 𝑘
36. 2x + 3yi = 6 + 2i
37. 3x + 2yi = 2(9 – 10i)
38. 𝑦 = √𝑥
40. 𝑦 = −√𝑥 + 2
Creating a
table can help
h is your right/left shift.
h is opposite!
Complex
Numbers
cont.
Solving
Radical
Equations
X
0
1
4
9
Y
k is your up/down shift.
a effects the curve shape
and –a “flips” graph.
Always do flips first!
21.
24.
7+√2
=
22.
25.
6  12 =
Do imaginary part first.
80 =
29.
27. (5 − 2𝑖)(4 − 𝑖) =
28.
2 6
=
4 6
3 2
3i
30.
2i
=
3i
3
6−𝑖 √5
=
=
33. √𝑥 2 + 4 − 2 = 𝑥
32. √4𝑟 − 6 = √𝑟
39. 𝑦 = √𝑥 − 2 + 1