2 rules for radicals

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.


a
a
=
b
b
EXAMPLE # 1 :


18
18
=
= 9 =3
2
2


a
a
=
b
b
EXAMPLE # 1 :

a b = a ⋅b


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.


a
a
=
b
b
EXAMPLE # 1 :

a b = a ⋅b
EXAMPLE # 2 :


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


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


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

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


n

a =a
m

EXAMPLE # 4 :

m
n

a
a
=
b
b
7 a 2b 3 ⋅ 7 a 6b1

a b = ab


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…


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


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…


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…


n

a =a
m

EXAMPLE # 5 :

a
a
=
b
b

m
n

24m 6 n 9
6m 2 n 3

a b = ab


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


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…

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.

same.
EXAMPLE :

4

36a 6b8c12

same.
EXAMPLE :

4

36a 6b8c12
1
4

6
4

8
4

= 36 a b c

First, rewrite using rational exponents….
12
4

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

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

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

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

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

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

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…

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 ?

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

One with a remainder of one.

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

One with a remainder of one.
The integer that divides it goes outside, the remainder goes
inside…

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

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….

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

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

Any proper fraction goes under the radical.
To do this, apply the rational exponent rule backwards.
Only your exponent will appear…

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

2 with a remainder of 1…

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…

2 rules for radicals

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to 2 rules for radicals

Similar to 2 rules for radicals (20)

More from Nikki Wilkinson

More from Nikki Wilkinson (20)

Recently uploaded

Recently uploaded (20)

2 rules for radicals