SECTION 8.6 (CONT.)
Solving Equations with Radicals
Solving an Equation with Radicals
• Ex:
3
𝑝+5=
3
2𝑝 − 4
What went wrong???
3𝑥 + 4 = 8 + 𝑥
3𝑥 + 4 = 64 + 𝑥 2
⋮
What went wrong???
5𝑥 − 6 − 𝑥 + 3 = 3
5𝑥 − 6 − 𝑥 + 3 = 9
⋮
Example 1
Solve.
3𝑝 + 4 − 2𝑝 − 4 = 2
Example 2
Solve.
3
𝑥2
+ 5𝑥 + 1 =
3
𝑥 2 + 4𝑥
Example 3
Solve.
4
8𝑧 − 3 + 2 = 0
Example 4
Solve.
3
𝑟+1+5=4
SECTION 8.7
𝒊
Example 1
Write each as a product of a real number and 𝑖.
a)
−225
b)
−21
c)
−48
Example 2
Multiply.
a)
−9 ∙ −81
b)
−10 ∙ 2
Example 3
Divide.
a)
−300
−100
− −64
b)
−16
The Complex Numbers
Example 4
Add or subtract. Write your answer in 𝑎
a)
b)
+ 𝑏𝑖
9 + 𝑖 − (3 + 2𝑖)
7 + 2𝑖 + −4 − 𝑖
− (2 + 5𝑖)
form.
Example 5
Multiply.
a)
−8𝑖 (−2𝑖)
b) (7 − 2𝑖)(3 + 𝑖)
Example 6
Multiply.
5 − 𝑖 (5 + 𝑖) 2 − 𝑖
2
Rationalizing the Denominator (cont.)
• If we have a complex number in our denominator,
we need to multiply the numerator and
denominator by the complex conjugate of the
denominator.
Expression
𝑎 + 𝑏𝑖
3 − 7𝑖
4𝑖
Conjugate
Example 7
Divide.
8𝑖
2 + 2𝑖
Example 8
Divide.
3−𝑖
−𝑖
Powers of 𝒊
1
𝑖 =
𝑖2 =
𝑖3 =
𝑖4 =
𝑖5 =
6
𝑖 =
7
𝑖 =
𝑖8 =
⋮
Example 9
Find each power of 𝑖.
a) 𝑖 48
b) 𝑖
83
c) 𝑖
−17