Download UNIT 2 WORKSHEET 12 RADICALS REVIEW PACKET

UNIT 2 WORKSHEET 12
RADICALS REVIEW PACKET
Simplify each of the following. Remember to use absolute value symbols when needed.
A)
E)
3
36x8 y 4
B)
64x3 y 6
F)
3
54x 4 y 2
C)
81( 3 x + 1)
4
G)
4
( 3x )
D)
x 4 y −6
H)
2 4
3
16 x5
27 x 2
36 x 2 − 36 y 2
Multiplication with Radicals
Simplify each of the following.
A) 2 2 ⋅ 4 5
12 x 3 ⋅ 18 x 2 ⋅ 27 x 2
D)
G)
(
2− 7
)
(2 2 )
6
B) 6 3 ⋅ 2 2 ⋅ 6
C)
E) 7 42 x 4 y 3 ⋅ 2 56 x5 y 3
F) 3 2 4 5 + 6
2
H)
(4
3 −2 5
)
2
(
(
I) 5 2 − 3
)
2
)
Rationalizing and Dividing Radicals
When working with radicals, a radical cannot be in the denominator. When left with a radical
in the denominator, the expression must be rationalized. Multiply the top and bottom of the
fraction by what is needed, not solely by what’s in the radical. This will come into play when
dividing radicals. When faced with a binomial in the denominator, the top and bottom must
be multiplied by the conjugate.
Rationalize the denominator of each of the following.
1
5
8
A)
B)
C) 5
3
10
12
5
E)
3
I)
15x 3 y 6 z 2
10
F)
5
30
7+ 2
J)
4
G)
16x 4 y 8
D)
3
H)
12x5 y 7
12
5− 6
K)
3
12x3 y 2
3
2
20
3 2 −4
Addition/Subtraction with Radicals
Simplify each of the following.
A) 6 3 + 7 3
B) 4 2 − 9 2 + 6 2
C)
D) 9 48 − 3 27 + 12 147
E) 13 90 − 4 40 + 7 250
F) 12 3 24 − 6 3 81
12 + 27
Equations with Radicals
When solving an equation with radicals, always check the solutions for any extraneous roots.
Any solution that does not make the original equation true is an extraneous root.
Find all real solutions for each of the following equations. Be sure to check for extraneous roots.
A) 2 x − 3 = 1
B)
2x = 4
C)
x −7 = 0
D)
x +6=0
E)
x +5 = 3
F)
x+6 − x = 2
G)
3x + 1 − 2 x − 1 = 1
H) 7 + 21 − 3x = x
I)
x + 33 − x = 3
3
Imaginary and Complex Numbers
Since the square root of a negative number is not real, a different type of number was invented
to represent them. Imaginary numbers come from taking the square root of a negative
number. The following is useful when dealing with imaginary numbers.
i = −1 = i
i2 =
(
−1
)
2
i 5 = i ⋅ i 4 = i ⋅ (1) = i
i 6 = i 2 ⋅ i 4 = ( −1)(1) = −1
= −1
i 3 = i ⋅ i 2 = i ⋅ ( −1) = −i
i 7 = i 3 ⋅ i 4 = ( −i )(1) = −i
i 4 = i 2 ⋅ i 2 = ( −1)( −1) = 1
i 8 = i 4 ⋅ i 4 = (1) ⋅ (1) = 1
This pattern will always repeat and continue. When asked to simplify something like i 22 , just
divide 22 by 4. The remainder is 2. The reason this is done is that for every power that is a
multiple of 4, the result is one (see above examples). The next step is to evaluate i 2 . Here is
what is actually happening.
i 22 = i 20 ⋅ i 2 = ( i 4 ) ⋅ i 2 = (1) ⋅ i 2 = i 2 = −1
5
5
Using the power of a power rule, the term i 20 can be rewritten as
(i ) .
4 5
The term i 4 is used
inside the parenthesis because i 4 = 1 .
Evaluate and simplify each of the following.
−12
C)
−8
G) 4 −16
H) − −54
J) 3i 2
K) −12i17
L) 6i 254
N) i 417
O) 17i 2436
P) −i 513
A)
−9
B)
E)
−121
F)
I)
−75
M) 16i 218
3
3
−27
D)
−49
Operations with Complex Numbers
A complex number is a real number and an imaginary number put together. Complex
numbers always come in the standard form a + bi . When solving a problem or evaluating an
expression where the solution is a complex number, the solution must be written in standard
form.
Evaluate each of the following expressions.
A) 2 ( 3 − 4i ) − 6 ( 8 + 2i )
B) 6 − ( −4 + 3i ) − ( −8 − 12i )
C) 15i − (12 − 10i )
−144 − −81
E) 2 −81 + −121 − −9
F) 3 −18 + 4 −32
D)
G)
−8 + −32
12
J) ( 5 − 3i )( 2 − 6i )
M)
( 3 + 4i )( 3 − 4i )
H)
15 − −27
18
K) 4 ( 5 − 3i ) − ( 2 − i )
N)
( 6 − 5i )( 6 + 5i )
I)
2
−12 − −28
32
L) ( 5 − 12i )
O)
(
2
−12 + 3
)
2
Division with Complex Numbers
Division with complex numbers is much like rationalizing a denominator. You cannot have a
complex number in the denominator, so multiply top and bottom by the conjugate. Remember,
your answer must be written in standard form.
Divide each of the following.
A)
2
3−i
B)
8i
4 − 3i
C)
15
2−i
D)
10
1 + 7i
E)
3+i
3−i
F)
3 − 4i
4 + 3i
G)
3 − 5i
3 + 5i
H)
5 − 3i
2 + 3i
More Equations
Solve each of the following equations.
A) x 2 + 400 = 0
B) 4 x 2 + 39 = 7
C) 5 x 2 = −20
D) 2 x 2 + 90 = 0
E) 3 x 2 + 14 = 8
F) 3 x 2 + 48 = 0