Important Formulas: Exponents and Radicals | Mathematics for Digital SAT PDF Download

 Page 1


Formulas for Exponent and Radicals
Algebraic Rules for Manipulating Exponential and Radicals Expressions.
In the following, n;m;k;j are arbitrary -
. they can be integers or rationals or real numbers.
b
n

b
m
b
k
=b
n+mk
Add exponents in the numerator and
Subtract exponent in denominator.

a
n

b
m
c
k

j
=
a
n
j

b
m
j
c
k
j
The exponent outside the parentheses
Multiplies the exponents inside.

a
n
b
m

1
=
b
m
a
n
Negative exponent "
ips" a fraction.
b
0
= 1 b =b
1
Don't forget these
Convert Radicals to Exponent notation
p
a =a
1=2
m
p
a =a
1=m
m
p
a
n
=a
n=m
Radicals - Reducing
p
a
2

b =a
p
b Remove squares from inside
m
p
a
m

b =a
m
p
b
Exponent and Radicals - Solving Equations
Solve a power by a root
x
n=m
=y,x =y
m=n
Solve a root by a power
1
Page 2


Formulas for Exponent and Radicals
Algebraic Rules for Manipulating Exponential and Radicals Expressions.
In the following, n;m;k;j are arbitrary -
. they can be integers or rationals or real numbers.
b
n

b
m
b
k
=b
n+mk
Add exponents in the numerator and
Subtract exponent in denominator.

a
n

b
m
c
k

j
=
a
n
j

b
m
j
c
k
j
The exponent outside the parentheses
Multiplies the exponents inside.

a
n
b
m

1
=
b
m
a
n
Negative exponent "
ips" a fraction.
b
0
= 1 b =b
1
Don't forget these
Convert Radicals to Exponent notation
p
a =a
1=2
m
p
a =a
1=m
m
p
a
n
=a
n=m
Radicals - Reducing
p
a
2

b =a
p
b Remove squares from inside
m
p
a
m

b =a
m
p
b
Exponent and Radicals - Solving Equations
Solve a power by a root
x
n=m
=y,x =y
m=n
Solve a root by a power
1
Example
a) Simplify

2
5

3
Method

2
5

3
=
2
3
5
3
=
2
 2
 2
5
 5
 5
=
8
125
b) Simplify

2
 3
2
5
3

2
Method

2
 3
2
5
3

2
=
2
2

 3
2
2
5
3
2
=
4
 81
15; 625
=
324
15;625
Illustration: where is the negative?
c) Simplify

 3


4
( the 'negative' is inside the parentheses)
Method

 3


4
= (3)
 (3)
 (3)
 (3) =81
d) Simplify

3


4
( the 'negative' is outside the parentheses)
Method


3


4
=(3)
 (3)
 (3)
 (3) =81
e) Simplify

2
5

3
( the 'negative' is in the exponent)
Method

2
5

3
=
1
(2=5)
3
= (or =

5
3

3
) =
5
3
2
3
=
125
8
2
Page 3


Formulas for Exponent and Radicals
Algebraic Rules for Manipulating Exponential and Radicals Expressions.
In the following, n;m;k;j are arbitrary -
. they can be integers or rationals or real numbers.
b
n

b
m
b
k
=b
n+mk
Add exponents in the numerator and
Subtract exponent in denominator.

a
n

b
m
c
k

j
=
a
n
j

b
m
j
c
k
j
The exponent outside the parentheses
Multiplies the exponents inside.

a
n
b
m

1
=
b
m
a
n
Negative exponent "
ips" a fraction.
b
0
= 1 b =b
1
Don't forget these
Convert Radicals to Exponent notation
p
a =a
1=2
m
p
a =a
1=m
m
p
a
n
=a
n=m
Radicals - Reducing
p
a
2

b =a
p
b Remove squares from inside
m
p
a
m

b =a
m
p
b
Exponent and Radicals - Solving Equations
Solve a power by a root
x
n=m
=y,x =y
m=n
Solve a root by a power
1
Example
a) Simplify

2
5

3
Method

2
5

3
=
2
3
5
3
=
2
 2
 2
5
 5
 5
=
8
125
b) Simplify

2
 3
2
5
3

2
Method

2
 3
2
5
3

2
=
2
2

 3
2
2
5
3
2
=
4
 81
15; 625
=
324
15;625
Illustration: where is the negative?
c) Simplify

 3


4
( the 'negative' is inside the parentheses)
Method

 3


4
= (3)
 (3)
 (3)
 (3) =81
d) Simplify

3


4
( the 'negative' is outside the parentheses)
Method


3


4
=(3)
 (3)
 (3)
 (3) =81
e) Simplify

2
5

3
( the 'negative' is in the exponent)
Method

2
5

3
=
1
(2=5)
3
= (or =

5
3

3
) =
5
3
2
3
=
125
8
2
More Examples
f) Simplify
x
3

x
7
x
5
Method
x
3

x
7
x
5
=x
3+75
=x
5
g) Simplify (2a
3
b
2
)(3ab
4
)
3
Method
(2a
3
b
2
)(3ab
4
)
3
= 2a
3
b
2

 3
3
a
3
b
4
3
= (2
 27)(a
3+3
)(b
2+12
)
= 54a
6
b
14
h) Simplify

x
y

3

y
2
x
z

4
(give answer with only positive exponents )
Method

x
y

3

y
2
x
z

4
=
x
3
y
3


y
2
4
x
4
z
4
=
x
3+4
y
83
z
4
=
x
7
y
5
z
4
3
Page 4


Formulas for Exponent and Radicals
Algebraic Rules for Manipulating Exponential and Radicals Expressions.
In the following, n;m;k;j are arbitrary -
. they can be integers or rationals or real numbers.
b
n

b
m
b
k
=b
n+mk
Add exponents in the numerator and
Subtract exponent in denominator.

a
n

b
m
c
k

j
=
a
n
j

b
m
j
c
k
j
The exponent outside the parentheses
Multiplies the exponents inside.

a
n
b
m

1
=
b
m
a
n
Negative exponent "
ips" a fraction.
b
0
= 1 b =b
1
Don't forget these
Convert Radicals to Exponent notation
p
a =a
1=2
m
p
a =a
1=m
m
p
a
n
=a
n=m
Radicals - Reducing
p
a
2

b =a
p
b Remove squares from inside
m
p
a
m

b =a
m
p
b
Exponent and Radicals - Solving Equations
Solve a power by a root
x
n=m
=y,x =y
m=n
Solve a root by a power
1
Example
a) Simplify

2
5

3
Method

2
5

3
=
2
3
5
3
=
2
 2
 2
5
 5
 5
=
8
125
b) Simplify

2
 3
2
5
3

2
Method

2
 3
2
5
3

2
=
2
2

 3
2
2
5
3
2
=
4
 81
15; 625
=
324
15;625
Illustration: where is the negative?
c) Simplify

 3


4
( the 'negative' is inside the parentheses)
Method

 3


4
= (3)
 (3)
 (3)
 (3) =81
d) Simplify

3


4
( the 'negative' is outside the parentheses)
Method


3


4
=(3)
 (3)
 (3)
 (3) =81
e) Simplify

2
5

3
( the 'negative' is in the exponent)
Method

2
5

3
=
1
(2=5)
3
= (or =

5
3

3
) =
5
3
2
3
=
125
8
2
More Examples
f) Simplify
x
3

x
7
x
5
Method
x
3

x
7
x
5
=x
3+75
=x
5
g) Simplify (2a
3
b
2
)(3ab
4
)
3
Method
(2a
3
b
2
)(3ab
4
)
3
= 2a
3
b
2

 3
3
a
3
b
4
3
= (2
 27)(a
3+3
)(b
2+12
)
= 54a
6
b
14
h) Simplify

x
y

3

y
2
x
z

4
(give answer with only positive exponents )
Method

x
y

3

y
2
x
z

4
=
x
3
y
3


y
2
4
x
4
z
4
=
x
3+4
y
83
z
4
=
x
7
y
5
z
4
3
More Examples with negatives
i) Simplify
6st
4
2s
2
t
2
(give answer with only positive exponents )
Negative exponents 
ip location: A negative exponent in the numerator
moves to the denominator. And a negative exponent in the denominator
moves to the numerator.
Method
6st
4
2s
2
t
2
=
6ss
2
2t
4
t
2
=
3s
3
t
6
j) Simplify

y
3z
3

2
(give answer with only positive exponents )
A Negative exponent '
ips' the fraction.
Method

y
3z
3

2
=

3z
3
y

2
=
9z
6
y
2
4
Page 5


Formulas for Exponent and Radicals
Algebraic Rules for Manipulating Exponential and Radicals Expressions.
In the following, n;m;k;j are arbitrary -
. they can be integers or rationals or real numbers.
b
n

b
m
b
k
=b
n+mk
Add exponents in the numerator and
Subtract exponent in denominator.

a
n

b
m
c
k

j
=
a
n
j

b
m
j
c
k
j
The exponent outside the parentheses
Multiplies the exponents inside.

a
n
b
m

1
=
b
m
a
n
Negative exponent "
ips" a fraction.
b
0
= 1 b =b
1
Don't forget these
Convert Radicals to Exponent notation
p
a =a
1=2
m
p
a =a
1=m
m
p
a
n
=a
n=m
Radicals - Reducing
p
a
2

b =a
p
b Remove squares from inside
m
p
a
m

b =a
m
p
b
Exponent and Radicals - Solving Equations
Solve a power by a root
x
n=m
=y,x =y
m=n
Solve a root by a power
1
Example
a) Simplify

2
5

3
Method

2
5

3
=
2
3
5
3
=
2
 2
 2
5
 5
 5
=
8
125
b) Simplify

2
 3
2
5
3

2
Method

2
 3
2
5
3

2
=
2
2

 3
2
2
5
3
2
=
4
 81
15; 625
=
324
15;625
Illustration: where is the negative?
c) Simplify

 3


4
( the 'negative' is inside the parentheses)
Method

 3


4
= (3)
 (3)
 (3)
 (3) =81
d) Simplify

3


4
( the 'negative' is outside the parentheses)
Method


3


4
=(3)
 (3)
 (3)
 (3) =81
e) Simplify

2
5

3
( the 'negative' is in the exponent)
Method

2
5

3
=
1
(2=5)
3
= (or =

5
3

3
) =
5
3
2
3
=
125
8
2
More Examples
f) Simplify
x
3

x
7
x
5
Method
x
3

x
7
x
5
=x
3+75
=x
5
g) Simplify (2a
3
b
2
)(3ab
4
)
3
Method
(2a
3
b
2
)(3ab
4
)
3
= 2a
3
b
2

 3
3
a
3
b
4
3
= (2
 27)(a
3+3
)(b
2+12
)
= 54a
6
b
14
h) Simplify

x
y

3

y
2
x
z

4
(give answer with only positive exponents )
Method

x
y

3

y
2
x
z

4
=
x
3
y
3


y
2
4
x
4
z
4
=
x
3+4
y
83
z
4
=
x
7
y
5
z
4
3
More Examples with negatives
i) Simplify
6st
4
2s
2
t
2
(give answer with only positive exponents )
Negative exponents 
ip location: A negative exponent in the numerator
moves to the denominator. And a negative exponent in the denominator
moves to the numerator.
Method
6st
4
2s
2
t
2
=
6ss
2
2t
4
t
2
=
3s
3
t
6
j) Simplify

y
3z
3

2
(give answer with only positive exponents )
A Negative exponent '
ips' the fraction.
Method

y
3z
3

2
=

3z
3
y

2
=
9z
6
y
2
4
More Examples
k) Simplify
(2x
3
)
2
(3x
4
)
(x
3
)
4
Method
(2x
3
)
2
(3x
4
)
(x
3
)
4
=
2
2
x
3
2

 3x
4
x
3
4
=
(4
 3)x
6
x
4
x
12
= 12x
6+412
=
12
x
2
5