U
n
i
t
1
–
N
um
be
rs
,
E
xpr
e
ss
i
ons
&
F
un
c
t
i
ons
—
F
u
ll De
pt
h S
tu
d
y
G
u
ide
(
NC Ma
t
h
2
CDM
)
1
.
REAL NUMBERS
—
Dee
p
Di
v
e
Ra
t
i
on
al
vs
.
I
rr
a
t
i
on
al
Ra
t
i
on
al
:
ca
n
be
wr
i
tt
e
n
a
s
a f
r
ac
t
i
on
o
f
two
i
nt
ege
rs
(
a
/
b
).
I
n
cl
u
de
s
i
nt
ege
rs
,
t
e
rm
i
n
a
t
i
n
g deci
m
al
s
,
a
n
d
r
e
p
ea
t
i
n
g deci
m
al
s
.
E
x
a
mp
le
s
:
4
,
-
3
,
1
/
2
,
0
.
75
,
0
.
333
...
I
rr
a
t
i
on
al
:
CANNOT be
wr
i
tt
e
n
a
s
a f
r
ac
t
i
on
—
deci
m
al g
o
e
s
f
or
e
v
e
r
w
i
t
h
no
r
e
p
ea
t
i
n
g
p
a
tt
e
rn
.
E
x
a
mp
le
s
:
π
,
√
2
,
√
3
,
e
.
I
mport
a
nt
tr
a
p
:
√
(
p
e
r
fec
t
squ
a
r
e
)
i
s
ac
tu
all
y
r
a
t
i
on
al
(
√
16
=
4
),
so
al
w
a
ys
s
i
mp
lif
y
a
r
adical bef
or
e decidi
n
g
.
Si
mp
lif
y
Radical
s
P
ro
ce
ss
:
fin
d
t
he la
r
ge
st
p
e
r
fec
t
squ
a
r
e fac
tor
o
f
t
he
num
be
r
,
pu
ll i
t
out
.
√
72
=
√
(
36
×
2
)
=
√
36
×
√
2
=
6
√
2
Wi
t
h
v
a
r
iable
s
:
pu
ll
out
p
ai
rs
o
f
m
a
t
chi
n
g fac
tors
.
√
(
x⁵
)
=
√
(
x⁴
×
x
)
=
x²
√
x
Si
mp
lif
y
C
u
be R
oots
Sa
m
e idea
,
b
ut
l
oo
k f
or
PERFECT CUBE fac
tors
(
8
,
27
,
64
,
125
...)
i
nst
ead
o
f
p
e
r
fec
t
squ
a
r
e
s
.
∛
54
=
∛
(
27
×
2
)
=
∛
27
×
∛
2
=
3
∛
2
Wi
t
h
v
a
r
iable
s
,
pu
ll
out
g
roups
o
f THREE
m
a
t
chi
n
g fac
tors
:
∛
(
x⁷
)
=
∛
(
x⁶
×
x
)
=
x²
∛
x
C
onv
e
rt
Be
tw
ee
n
Radical
s
a
n
d Ra
t
i
on
al E
xpon
e
nts
R
u
le
:
ⁿ
√
(
xᵐ
)
=
x
^(
m
/
n
)
The
root
i
n
de
x
(
n
)
bec
om
e
s
t
he DENOMINATOR
o
f
t
he e
xpon
e
nt
.
The
pow
e
r
i
ns
ide
(
m
)
bec
om
e
s
t
he NUMERATOR
.
E
x
a
mp
le
:
√
(
x³
)
=
x
^(
3
/
2
)
E
x
a
mp
le
:
∛
(
x⁵
)
=
x
^(
5
/
3
)
E
x
a
mp
le
(
r
e
v
e
rs
e
):
x
^(
2
/
3
)
=
∛
(
x²
)
A
pp
l
y
E
xpon
e
nt
R
u
le
s
R
u
le F
ormu
la
P
ro
d
u
c
t
o
f
pow
e
rs xᵃ
·
xᵇ
=
x
^(
a
+
b
)
Q
uot
ie
nt
o
f
pow
e
rs xᵃ
/
xᵇ
=
x
^(
a
−
b
)
R
u
le F
ormu
la
P
ow
e
r
o
f a
pow
e
r
(
xᵃ
)
ᵇ
=
x
^(
ab
)
P
ow
e
r
o
f a
pro
d
u
c
t
(
xy
)
ᵃ
=
xᵃyᵃ
Nega
t
i
v
e e
xpon
e
nt x
⁻
ᵃ
=
1
/
xᵃ
Ze
ro
e
xpon
e
nt x⁰
=
1
(
x
≠
0
)
Si
mp
lif
y
E
xpr
e
ss
i
ons
w
i
t
h E
xpon
e
nts
A
pp
l
y
ru
le
s
i
n
c
om
bi
n
a
t
i
on
,
wor
ki
n
g
st
e
p
b
y
st
e
p
,
a
n
d al
w
a
ys
lea
v
e a
nsw
e
rs
w
i
t
h POSITIVE
e
xpon
e
nts
.
E
x
a
mp
le
:
(
2x³y
⁻
²
)
²
=
4x⁶y
⁻
⁴
=
4x⁶
/
y⁴
2
.
COMPLEX NUMBERS
—
Dee
p
Di
v
e
i
=
√
(−
1
)
The i
m
agi
n
a
ry
un
i
t
i
s
de
fin
ed
so
t
ha
t
n
ega
t
i
v
e
num
be
rs
ca
n
ha
v
e
squ
a
r
e
roots
.
i
²
=
−
1
.
√
(−
9
)
=
√
9
·
√
(−
1
)
=
3
i
Si
mp
lif
y
P
ow
e
rs
o
f i
P
ow
e
rs
o
f i c
y
cle i
n
a
p
a
tt
e
rn
o
f
4
:
i
¹
=
i
,
i
²
=
−
1
,
i
³
=
−
i
,
i
⁴
=
1
,
t
he
n
i
t
r
e
p
ea
ts
.
Sh
ort
c
ut
:
di
v
ide
t
he e
xpon
e
nt
b
y
4
,
us
e
t
he REMAINDER
to
fin
d
w
hich
o
f
t
he
4
v
al
u
e
s
i
t
m
a
t
che
s
.
E
x
a
mp
le
:
i
²³
→
23
÷
4
=
5
r
e
m
ai
n
de
r
3
→
i
²³
=
i
³
=
−
i
Add a
n
d S
u
b
tr
ac
t
C
omp
le
x
N
um
be
rs
C
om
bi
n
e
r
eal
p
a
rts
to
ge
t
he
r
,
a
n
d i
m
agi
n
a
ry
p
a
rts
to
ge
t
he
r
(
like c
om
bi
n
i
n
g like
t
e
rms
).
(
3
+
4
i
)
+
(
2
−
5
i
)
=
(
3
+
2
)
+
(
4
−
5
)
i
=
5
−
i
M
u
l
t
i
p
l
y
C
omp
le
x
N
um
be
rs
U
s
e FOIL
,
t
he
n
s
i
mp
lif
y
us
i
n
g i
²
=
−
1
.
(
2
+
3
i
)(
1
−
4
i
)
=
2
−
8
i
+
3
i
−
12
i
²
=
2
−
5
i
−
12
(−
1
)
=
2
−
5
i
+
12
=
14
−
5
i
3
.
POLYNOMIALS
—
Dee
p
Di
v
e
Add
/
S
u
b
tr
ac
t
P
o
l
ynom
ial
s
C
om
bi
n
e LIKE TERMS
on
l
y
(
s
a
m
e
v
a
r
iable AND
s
a
m
e e
xpon
e
nt
).
F
or
su
b
tr
ac
t
i
on
,
di
str
ib
ut
e
t
he
n
ega
t
i
v
e
s
ig
n
ac
ross
e
v
e
ry
t
e
rm
o
f
t
he
s
ec
on
d
po
l
ynom
ial
first
.
(
3x²
+
5x
−
2
)
−
(
x²
−
4x
+
6
)
=
3x²
+
5x
−
2
−
x²
+
4x
−
6
=
2x²
+
9x
−
8
M
u
l
t
i
p
l
y
P
o
l
ynom
ial
s
Di
str
ib
ut
e e
v
e
ry
t
e
rm
o
f
t
he
first
po
l
ynom
ial ac
ross
e
v
e
ry
t
e
rm
o
f
t
he
s
ec
on
d
,
t
he
n
c
om
bi
n
e
like
t
e
rms
.
FOIL
(
f
or
two
bi
nom
ial
s
)
Fi
rst
,
O
ut
e
r
,
I
nn
e
r
,
La
st
.
(
x
+
3
)(
x
−
5
)
=
x²
−
5x
+
3x
−
15
=
x²
−
2x
−
15
Fac
tor
GCF
(
G
r
ea
t
e
st
C
ommon
Fac
tor
)
Fi
n
d
t
he la
r
ge
st
t
e
rm
(
num
be
r
a
n
d
/
or
v
a
r
iable
)
c
ommon
to
e
v
e
ry
t
e
rm
,
pu
ll i
t
out
f
ront
.
6x³
+
9x²
=
3x²
(
2x
+
3
)
Fac
tor
T
r
i
nom
ial
s
F
or
x²
+
b
x
+
c
:
fin
d
two
num
be
rs
t
ha
t
mu
l
t
i
p
l
y
to
c a
n
d add
to
b
.
F
or
a
x²
+
b
x
+
c
(
a
≠
1
):
us
e
g
roup
i
n
g
—
mu
l
t
i
p
l
y
a
·
c
,
fin
d
two
num
be
rs
mu
l
t
i
p
l
y
i
n
g
to
t
ha
t
a
n
d addi
n
g
to
b
,
sp
li
t
t
he
m
iddle
t
e
rm
,
fac
tor
b
y
g
roup
i
n
g
.
Fac
tor
Di
ff
e
r
e
n
ce
o
f S
qu
a
r
e
s
a
²
−
b
²
=
(
a
+
b
)(
a
−
b
)
E
x
a
mp
le
:
x²
−
25
=
(
x
+
5
)(
x
−
5
)
T
r
a
p
:
t
he SUM
o
f
squ
a
r
e
s
(
a
²
+
b
²
)
d
o
e
s
NOT fac
tor
ov
e
r
t
he
r
eal
s
.
Fac
tor
C
omp
le
t
el
y
Al
w
a
ys
st
a
rt
b
y
pu
lli
n
g
out
t
he GCF
first
,
THEN l
oo
k f
or
f
urt
he
r
fac
tor
i
n
g
(
tr
i
nom
ial
,
di
ff
e
r
e
n
ce
o
f
squ
a
r
e
s
,
e
t
c
.).
"
C
omp
le
t
el
y
"
m
ea
ns
kee
p
fac
tor
i
n
g
unt
il
not
hi
n
g fac
tors
f
urt
he
r
.
E
x
a
mp
le
:
2x³
−
8x
=
2x
(
x²
−
4
)
=
2x
(
x
+
2
)(
x
−
2
)
4
.
LINEAR EQUATIONS
—
Dee
p
Di
v
e
O
n
e
-
S
t
e
p
E
qu
a
t
i
ons
U
n
d
o
on
e
op
e
r
a
t
i
on
(
add
/
su
b
tr
ac
t
or
mu
l
t
i
p
l
y
/
di
v
ide
)
to
i
so
la
t
e
t
he
v
a
r
iable
.
x
+
7
=
12
→
x
=
5
T
wo
-
S
t
e
p
E
qu
a
t
i
ons
U
n
d
o
addi
t
i
on
/
su
b
tr
ac
t
i
on
FIRST
,
t
he
n
mu
l
t
i
p
lica
t
i
on
/
di
v
i
s
i
on
.
3x
+
4
=
19
→
3x
=
15
→
x
=
5
End of Document
330
