The document discusses rules for simplifying radical expressions. It states that if the index of radicals in a division or multiplication expression are the same, the division can be rewritten as a division of the radicals and the multiplication can be rewritten as a multiplication of the radicals. Examples are provided to demonstrate simplifying expressions using these rules together with rules for rational exponents and reducing the index of fractional exponents.

1. Algebraic Expressions - Rules for Radicals
Let’s review some concepts from Algebra 1.
a
a
=
b
b
If you have the same index, you can rewrite division of
radical expressions to help simplify. This rule also
applies in the opposite direction.

2. Algebraic Expressions - Rules for Radicals
Let’s review some concepts from Algebra 1.
a
a
=
b
b
EXAMPLE # 1 :
If you have the same index, you can rewrite division of
radical expressions to help simplify. This rule also
applies in the opposite direction.
18
18
=
= 9 =3
2
2

3. Algebraic Expressions - Rules for Radicals
Let’s review some concepts from Algebra 1.
a
a
=
b
b
EXAMPLE # 1 :
a b = a ⋅b
If you have the same index, you can rewrite division of
radical expressions to help simplify. This rule also
applies in the opposite direction.
18
18
=
= 9 =3
2
2
If you have the same index, you can rewrite
multiplication of radical expressions to help simplify.
This rule also applies in the opposite direction.

4. Algebraic Expressions - Rules for Radicals
Let’s review some concepts from Algebra 1.
a
a
=
b
b
EXAMPLE # 1 :
a b = a ⋅b
EXAMPLE # 2 :
If you have the same index, you can rewrite division of
radical expressions to help simplify. This rule also
applies in the opposite direction.
18
18
=
= 9 =3
2
2
If you have the same index, you can rewrite
multiplication of radical expressions to help simplify.
This rule also applies in the opposite direction.
2 8 = 2 ⋅ 8 = 16 = 4

5. Algebraic Expressions - Rules for Radicals
n
a =a
m
m
n
This rule is used to change a radical expression into an
algebraic expression with a rational ( fraction ) exponent.
You divide the exponent by the index ( your root )
Remember, if no root is shown, the index = 2

6. Algebraic Expressions - Rules for Radicals
n
a =a
m
This rule is used to change a radical expression into an
algebraic expression with a rational ( fraction ) exponent.
m
n
You divide the exponent by the index ( your root )
Remember, if no root is shown, the index = 2
EXAMPLE # 3 :
3
x =x
2
2
3

7. Algebraic Expressions - Rules for Radicals
We are going to use these rules to simplify expressions. You
might have to use all of them together to simplify one problem.
n
a =a
m
m
n
a
a
=
b
b
a b = ab

8. Algebraic Expressions - Rules for Radicals
We are going to use these rules to simplify expressions. You
might have to use all of them together to simplify one problem.
n
a =a
m
EXAMPLE # 4 :
m
n
a
a
=
b
b
7 a 2b 3 ⋅ 7 a 6b1
a b = ab

9. Algebraic Expressions - Rules for Radicals
We are going to use these rules to simplify expressions. You
might have to use all of them together to simplify one problem.
n
a =a
m
EXAMPLE # 4 :
a
a
=
b
b
m
n
7 a 2b 3 ⋅ 7 a 6b1
= 49a 8b 4
a b = ab
Apply this rule first along with the
rule for multiplying variables…

10. Algebraic Expressions - Rules for Radicals
We are going to use these rules to simplify expressions. You
might have to use all of them together to simplify one problem.
n
a =a
m
EXAMPLE # 4 :
a
a
=
b
b
m
n
a b = ab
7 a 2b 3 ⋅ 7 a 6b1
= 49a 8b 4
= 49 ⋅ a 8 ⋅ b 4
Now lets split up each term so you
can see it…it’s the rule in the
opposite direction

11. Algebraic Expressions - Rules for Radicals
We are going to use these rules to simplify expressions. You
might have to use all of them together to simplify one problem.
n
a =a
m
EXAMPLE # 4 :
a
a
=
b
b
m
n
a b = ab
7 a 2b 3 ⋅ 7 a 6b1
= 49a 8b 4
= 49 ⋅ a 8 ⋅ b 4
8
2
4
2
= 7 a b = 7 a 4b 2
Find the square root of the integer
and apply this rule to your
variables…no index is shown so it
equals two…

12. Algebraic Expressions - Rules for Radicals
We are going to use these rules to simplify expressions. You
might have to use all of them together to simplify one problem.
n
a =a
m
EXAMPLE # 4 :
a
a
=
b
b
m
n
a b = ab
7 a 2b 3 ⋅ 7 a 6b1
= 49a 8b 4
= 49 ⋅ a 8 ⋅ b 4
8
2
4
2
= 7 a b = 7 a 4b 2
The middle steps are not necessary
if you want to mentally go from step
one to the answer…

13. Algebraic Expressions - Rules for Radicals
We are going to use these rules to simplify expressions. You
might have to use all of them together to simplify one problem.
n
a =a
m
EXAMPLE # 5 :
a
a
=
b
b
m
n
24m 6 n 9
6m 2 n 3
a b = ab

14. Algebraic Expressions - Rules for Radicals
We are going to use these rules to simplify expressions. You
might have to use all of them together to simplify one problem.
n
a =a
m
EXAMPLE # 5 :
a
a
=
b
b
m
n
24m 6 n 9
6m 2 n 3
a b = ab
Apply this rule first along with the
rule for dividing variables and
simplify under the radical…
24m 6 n 9
=
= 4m 4 n 6
6m 2 n 3

15. Algebraic Expressions - Rules for Radicals
We are going to use these rules to simplify expressions. You
might have to use all of them together to simplify one problem.
n
a =a
m
EXAMPLE # 5 :
a
a
=
b
b
m
n
a b = ab
24m 6 n 9
6m 2 n 3
24m 6 n 9
=
= 4m 4 n 6
6m 2 n 3
4
2
6
2
= 2m n = 2m n
2
3
Now apply the index rule…

16. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.

17. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
4
36a 6b8c12

18. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
4
36a 6b8c12
1
4
6
4
8
4
= 36 a b c
First, rewrite using rational exponents….
12
4

19. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
4
36a 6b8c12
1
4
6
4
8
4
12
4
1
4
3
2
= 36 a b c = 36 a b c
2 3
As you can see, the index of the first and
second term are now different…this is not
allowed

20. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
4
36a 6b8c12
6
4
1
4
8
4
12
4
1
4
3
2
= 36 a b c = 36 a b c
( )
= 6
2
1
4
2 3
3
2
a b 2c3
Rewrite 36 as 6 squared…when you apply
the rule for an exponent inside raised to an
exponent outside, the index will reduce…
(6 )
2
1
4
=6
2⋅ 1
4
=6
1
2

21. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
4
36a 6b8c12
6
4
1
4
8
4
12
4
3
2
1
4
= 36 a b c = 36 a b c
( )
= 6
2
1
4
3
2
2 3
3
2
1
2
a b c = 6 a b 2c3
2 3
Insert this into our problem…
(6 )
2
1
4
=6
2⋅ 1
4
=6
1
2

22. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
4
36a 6b8c12
6
4
1
4
8
4
12
4
3
2
1
4
= 36 a b c = 36 a b c
( )
= 6
2
1
4
3
2
1
2
3
2
2 3
a b c = 6 a b 2c3
2 3

23. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
4
36a 6b8c12
6
4
1
4
8
4
12
4
3
2
1
4
= 36 a b c = 36 a b c
( )
= 6
2
1
4
3
2
1
2
2 3
3
2
a b c = 6 a b 2c3
2 3
Writing the answer in radical form :
1. Any integer exponents appear outside the radical
2. Any proper fraction goes under the radical
3. Improper fractions are broken up into an integer
and a remainder. The integer part goes outside the
radical, the remainder goes inside the radical

24. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
4
36a 6b8c12
6
4
1
4
8
4
12
4
3
2
1
4
= 36 a b c = 36 a b c
( )
= 6
2
1
4
3
2
1
2
2 3
3
2
a b c = 6 a b 2c 3 = b 2c 3
2 3
Any integer exponents appear outside the radical

25. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
4
36a 6b8c12
6
4
1
4
8
4
12
4
3
2
1
4
= 36 a b c = 36 a b c
( )
= 6
2
1
4
3
2
1
2
2 3
3
2
a b c = 6 a b 2 c 3 = b 2 c 3 61
2 3
Any proper fraction goes under the radical.
To do this, apply the rational exponent rule backwards.
Only your exponent will appear…

26. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
4
36a 6b8c12
6
4
1
4
8
4
12
4
3
2
1
4
= 36 a b c = 36 a b c
( )
= 6
2
1
4
3
2
1
2
2 3
3
2
a b c = 6 a b 2c 3 = b 2c 3 6
2 3
Improper fractions are broken up into an integer and a
remainder. The integer part goes outside the radical, the
remainder goes inside the radical.
How many 2’s divide 3 with out going over ?

27. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
4
36a 6b8c12
6
4
1
4
8
4
12
4
3
2
1
4
= 36 a b c = 36 a b c
( )
= 6
2
1
4
3
2
1
2
2 3
3
2
a b c = 6 a b 2c 3 = b 2c 3 6
2 3
Improper fractions are broken up into an integer and a
remainder. The integer part goes outside the radical, the
remainder goes inside the radical.
How many 2’s divide 3 with out going over ?
One with a remainder of one.

28. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
4
36a 6b8c12
6
4
1
4
8
4
12
4
3
2
1
4
= 36 a b c = 36 a b c
( )
= 6
2
1
4
3
2
1
2
2 3
3
2
a b c = 6 a b 2 c 3 = b 2 c 3 a1 6 a1
2 3
Improper fractions are broken up into an integer and a
remainder. The integer part goes outside the radical, the
remainder goes inside the radical.
How many 2’s divide 3 with out going over ?
One with a remainder of one.
The integer that divides it goes outside, the remainder goes
inside…

29. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
6
First, rewrite using rational exponents and
reduce any fractional exponents….
8 x 6 y15 z 3
1
6
6
6
15
6
3
6
1
6
5
2
=8 x y z =8 x y z
1
1
2

30. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
6
8 x 6 y15 z 3
6
6
1
6
15
6
3
6
1
6
5
2
=8 x y z =8 x y z
( )
= 2
3
1
6
5
2
1
2
1
1
2
1
2
5
2
x y z =2 x y z
1
1
1
2
Now reduce the index of your integer….

31. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
6
8 x 6 y15 z 3
6
6
1
6
15
6
3
6
1
6
5
2
=8 x y z =8 x y z
= (2
3
)
1
6
5
2
1
2
1
1
2
1
2
5
2
1
2
x y z = 2 x y z = x1
1
1
Rewrite your answer in radical form….
Any integer exponents appear outside the radical

32. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
6
8 x 6 y15 z 3
6
6
1
6
15
6
3
6
1
6
5
2
=8 x y z =8 x y z
= (2
3
)
1
6
5
2
1
2
1
1
2
1
2
5
2
1
2
x y z = 2 x y z = x1 21 z1
1
1
Rewrite your answer in radical form….
Any proper fraction goes under the radical.
To do this, apply the rational exponent rule backwards.
Only your exponent will appear…

33. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
6
8 x 6 y15 z 3
6
6
1
6
15
6
3
6
1
6
5
2
=8 x y z =8 x y z
( )
= 2
3
1
6
5
2
1
2
1
1
2
1
2
5
2
1
2
x y z = 2 x y z = x1 y 2 21 z1 y1
1
1
Rewrite your answer in radical form….
Improper fractions are broken up into an integer and a
remainder. The integer part goes outside the radical, the
remainder goes inside the radical.
How many 2’s divide 5 with out going over ?
2 with a remainder of 1…

34. Algebraic Expressions - Rules for Radicals
Reducing the index – sometimes when applying the rule your fractional exponent
will change its index due to reducing. When that happens, all indexes must be the
same.
EXAMPLE :
6
8 x 6 y15 z 3
6
6
1
6
15
6
3
6
1
6
5
2
=8 x y z =8 x y z
= (2
3
)
1
6
5
2
1
2
1
1
2
1
2
5
2
1
2
x y z = 2 x y z = x1 y 2 21 z1 y1 = xy 2 2 zy
1
1
The ones as exponents are not needed in your
answer, I put them there so you could see
where things were going…